I am not going to list all of the mechanical differences between 4e and 5e; though that is the typical, and no doubt expected, answer to this question, 4e and 5e are simply much too different for that approach, in my opinion. Instead, this answer attempts to get at the extreme philosophical differences between 4e and 5e. Ultimately, you are going to have to treat 5e as its own separate product; knowledge of 4e will probably result more in bad habits or misplaced expectations, than it is to offer an inside track to learning 5e.
Now then, time for a history lesson:
D&D 4e was an attempt to start D&D over. It was rebuilt from the ground up, with lots of, for lack of a better term, “modern” design principles applied, and no sacred cow was safe.
D&D had accumulated a very large number of mechanics and other details over the years in a haphazard fashion, and many of them were considered either dubious ideas to begin with, or obsolete, meant for older playstyles that were no longer popular. Wizards thought it could do better by cleaning up the game, removing legacy content that was no longer fitting for the game they thought players wanted.
Market research suggested that D&D tables tended to focus on combat, tended to focus on big, epic narratives, and tended to be annoyed by bookkeeping, concerns about mechanics that didn’t work as advertised or that couldn’t keep up with other options, and how much work DM preparation required. They also found that people were sometimes reluctant to buy new material for their games, because DMs didn’t have the time or interest to vet it, and wouldn’t allow it in their games prior to doing so. As a company looking to sell books, this was obviously a problem.
So balance and ease of adding new material to a table was a major focus. WotC made the decision to label all books “core” and suggested that you ought to be able to show up to any table, anywhere, and use a character built with any of the published books.
To players, it was a promise that you could find a place to play that character you wanted to play (read: a promise that if you bought books from them, you’d be able to use them).
To DMs, it was a promise that you didn’t have to worry about the character built with some new book; as long as it was done by the book, it would work fine. They also promised digital tools to make checking that relatively simple.
In addition, D&D originally had a very strong logistical focus; accounting for all of your supplies, figuring out how to actually haul all of the dragon’s horde out after slaying it, and so on, were major parts of the game. Play preferences in the decades since D&D’s inception, however, had moved away from these, and tended to prefer to focus on characters and stories. There was also a greater embrace of “high fantasy,” not as concerned with perfectly simulating a world so much as simulating a certain narrative. So Wizards also sought to offer a lot of “quality of life” improvements to limit accounting and bookkeeping, to keep the game moving.
By the same token, embracing the “epic” “high fantasy” style meant embracing the extreme growth potential of the previous 3.5 edition, which had probably been accidental in that edition.
In order to accomplish these goals (and by and large, they did, though of course none of them were accomplished perfectly, or necessarily accomplished at all right out of the gate), Wizards changed a lot of things about the game. For examples:
Classes were streamlined, and to a casual review, homogenized, in the name of balance (and, presumably, to ease integration with the digital tools).
Everything revolved around the use of powers, discrete abilities that were basically supposed to sum up the totality of actions available to a character.
Roles, which were previously just an idea some players had, were made an explicit part of the rules of the game, and the game carefully protected classes’ roles, preventing classes from covering multiple roles too well.
Tiers weren’t precisely new – 3e had Epic as an optional post-20th ruleset, and the old BECMI rules are even named for their five tiers (Basic, Expert, Companion, Master, and Immortal) – but Epic rules were rarely used in 3e, and BECMI was quite old. Coming from AD&D (which had things called Epic, but did not involve more levels) or from one of the overwhelming majority of 3.x tables that never used the Epic rules, 4e had a much stronger and more explicit emphasis on becoming near-godlike. (This is despite the fact that the power available to 3.x characters at the high end absolutely dwarfs anything that 4e characters can do; that was an unintentional part of that system, and most tables never saw it.)
Many details that had previously required a lot of bookkeeping were glossed over, or simply ignored entirely in the favor of the epic narrative, especially if they weren’t relevant to combat.
Combat, which honestly had always constituted the largest chunk of D&D rules, became seen as very close to the only thing developers cared about. The little fiddly bits that were being glossed over were how D&D did non-combat, so with those gone, non-combat mechanics became quite abstract and steamlined
And working from the ground up, WotC eschewed quite a few mechanics that had a long legacy within D&D, the so-called “sacred cows.” Exactly which things were sacred varies based on who you asked, but since WotC was starting from scratch, a huge number of candidates for sacred cow status were not included in 4e.
The problem with all of this was that they changed a lot of things. A huge portion of the gaming tables could find something they hated about 4th edition, whether it was the (apparently) homogenous classes, the lack of detail in non-combat portions of the game, or the insistence that players should be able to show up using a new book, rather than DM acting as gatekeeper for new material. Or just whatever sacred cow they objected to the slaughter of; there were many.
In particular, 4e’s abstract and streamlined out-of-combat play, the lack of fiddly rules, was a problem. In many ways, this matched how prior editions of D&D were actually played – many, many tables ignored the fiddly rules and just did their own thing, out of combat – but the fact that the books didn’t get into details gave readers the idea that 4e was a game purely about combat, that they weren’t supposed to put much emphasis at all on non-combat aspects of the game. Removing the fiddly bits, even the ones that they didn’t use, made 4e “not real D&D” for a lot of people. And of course, many did use the fiddly bits, or at least used some of them.
This led to a fracturing of the D&D playerbase:
Quite a few players did move to 4e (and 4e did relatively well in terms of grabbing new players),
A large fraction of players stuck with the prior 3.5e (and many of those later moved to the spin-off by Paizo, Pathfinder).
Many even went back to the TSR-era D&D products, mostly 2nd edition, as 4e convinced them that they were not fans of WotC’s D&D wholesale (and just as with Pathfinder, an “old school revolution” developed to take advantage of that demand).
5e is an attempt to re-unite the player base. Rather than attempting to promise that all material will work at all tables, 5e has promised a massively modular design, allowing every table to tailor the game to exactly what they want. Many of the sacred cows, not included in 4e, have made a return. Where 4e was a highly detailed tactical simulator wherein almost all causes and effects had explicit rules, 5e relies massively on DM adjudication – just as previous editions had.
This is not without cost. 4e’s streamlining and balancing made it a very deep tactical combat game with an astonishing number of options. Being extremely detailed and explicit meant that the game gave players a lot of ability to anticipate how the various options they had would work, what would be more valuable. It also meant that DMs had a ton of guidance and resources to work with. One of the biggest advantages that 4e offered was the ease with which DMs could put together fun, interesting combat encounters that just worked – 5e is not in a position to guarantee that, and relies much more on DM work to make that happen.
Further, some of the quality-of-life improvements are gone, in the name of re-enshrining various sacred cows. Greater emphasis on these features is basically how non-4e editions of D&D do non-combat activity, but it means that there are a lot of mundane issues that 4e treated as trivial and did not spend player time on.
Finally, 5e does not have tiers, and is actually designed very strongly with being “flat” in mind. Bounded accuracy and other mechanics mean that even high-level characters can still be threatened by lower-level challenges, particularly if the latter have a numerical advantage. 20th-level 5e characters are not the “one-man army, about to start on a path to immortality” paragons that 20th-level 4e characters are.
So when moving from 4e to 5e, both players and DM will have to consider material more critically. Not all options are necessarily balanced with each other, or even relatively balanced, nor are they necessarily intended to be. Many things that the DM could simply look up in 4e, cannot be in 5e, and he will have to make something up (and that also means that players will not be able to know how many things are going to work until they ask their DM, and that knowledge most likely will apply only to that DM). Combat will also be much more simple, which will mean they are quicker, but may also mean they are less intrinsically interesting. Out of combat aspects of the game may take more of a focus, but much of that will still be left up to the DM (just more explicitly so, this time). In short, 4e is very much the odd one out among D&D editions, and 5e is a return to form. If 4e is the only D&D you have played, however, 5e may surprise, and even disappoint, you as much as 4e did for those who preferred the editions that came before.
Best Answer
I'm very familiar with 3.5, somewhat familiar with PF, but only know the core ruleset of 4e. Nevertheless, since no one else has answered, I thought I might as well take a stab at the major differences. I'd say the general theme is that PF offers more paths while building a character, at the cost of offering more dead ends as well. Similarly, monsters are more flexible, but more difficult for the DM to run.
I think you could write an interesting essay talking more about the philosophical and design differences between the two systems, but in answering this question it'll be more useful to address the observable differences point by point.
Class and monster roles
Fourth edition (4e) has clearly defined class and monster roles. This isn't the case at all in Pathfinder (PF). A fighter in PF might be good at dealing damage (striker) or a great tank (defender) but this will depend entirely on how you build your character. Monsters do not provide an obvious indication of how they should be used; the DM has to examine their abilities and think about what their role should be in battle.
It's very easy to build an ineffective character in PF, as a consequence of the greater flexibility you have in defining your role.
Multiclassing
Since in PF each class defines its own progression, every time a PC gains a level they can choose what class to advance in. Again, this means it is possible to design a character that performs very poorly. It would be a bit like creating a 4e character that drew from multiple classes, but was capable of using only heroic level powers while everyone else was choosing a paragon path.
Paths vs. Prestige Classes
In 4e, there are three distinct tiers of play, and you can choose a paragon path and epic destiny as you progress.
There's no direct analogue to the tiers in PF. The closest thing to a paragon path is a prestige class -- a class that can only be taken when you meet certain requirements. This was more of a big deal in 3.5, where it was used for character customization; in PF there are many fewer prestige classes, though they might be important for certain multiclass combinations.
Defenses v. Saves
This is purely technical, but confusing if you're not expecting it. Defenses in 4e are equivalent to saving throws in PF. If a wizard casts a spell, rather than rolling an attack, the defender rolls a saving throw against a DC set by the spell.
Automatic progressions
In 4e, you add half your level to a great many statistics. PF is not quite so straightforward -- your class(es) will determine your base attack bonus and saving throw progression, and you must pick how your skills improve every level. Other abilities like AC or initiative do not naturally increase as you level.
Powers, Spells, and Maneuvers
This is probably the most fundamental difference from a player's perspective.
In 4e, every class has a list of powers they can learn as they level. These powers are divided into at-will, daily, and encounter powers that dictate how often they can be used. Pretty much any special ability a player has is derived from a power -- and those powers are chiefly geared towards combat. Casting classes can gain rituals which are less combat related, and work off a different economy than combat powers.
In Pathfinder, it works a little differently. Casting classes have spells, which are probably closest in spirit to 4e powers, but tend to be more flexible. The number of spells you can choose from is quite large, and different classes have different mechanics for learning and casting these spells. Spells useful in combat consume exactly the same resources as more role-playing oriented spells.
Meanwhile, melee classes rarely have the same type of limited powers. Anyone can push, grab, pull, grapple, trip or disarm a foe during combat, and you can do so as often as you like. Doing so means sacrificing damage, though; you will only do damage with a regular attack. Unlike 4e, forced movement is quite difficult to pull off in PF, which is why a grid isn't as necessary.
As an example, a fighter in PF progresses chiefly because they get better at what they've been capable of doing all along, rather than getting spectacular new powers.
Monsters
This is a big difference from the DM's POV especially.
In addition to fitting certain roles (as mentioned above) monsters in 4e are often defined by a set of bespoke abilities. In contrast, monsters in PF always play by the same rules as the players. Their hp, attacks, saves, and skills are defined and limited by the type and HD of the monster. The majority of unique special abilities are copies of spells, or one of a small set of abilities common to many monsters. Often a powerful monster will have a large set of abilities they will never use in combat, but can provide role playing hooks.
There is no PF equivalent to elite or solo monsters -- you simply must use a higher level beast to provide a greater challenge to the party.
Overall the 4e system is much easier on the DM in creating encounters. For those so inclined, though, the PF system allows the creation of some interesting beasts; you can even add class levels to monsters, much like PCs can multiclass.
Magic items
The majority of magic items in PF do one of two things -- they either increase the attributes of a PC, or they can replicate the effects of spells. Magic weapons and armor combine several different attributes, almost Diablo style. Compared to the 4e items in core, they are much more varied, and what specific item your character has can drastically affect what they can do. The downside is that it becomes harder for a player to choose the 'correct' gear.