I have been running a Pathfinder campaign for two years and have used plenty of 3e and 3.5e adventures in it as well as native Pathfinder ones. (I've used multiple adventures from the Atlas Games Penumbra series, Green Ronin Freeport series, Goodman Games Wicked Fantasy Factory series, and older Paizo 3.5e APs and Pathfinder Modules.)
The short form is that 3e monsters/encounters count as CR (or EL)-2, and 3.5e ones count as CR (or EL)-1. In other words, you need a CR 8 3e monster or NPC to provide the punch a CR 6 one does in Pathfinder.
Besides that, you can do as much conversion as you want, but you really don't have to do any. Just run adventures higher in the range band - "D&D 3e for levels 3-5" means use on level 3 characters max unless you want to do conversion work.
My general process when using old ed adventures and actually converting is:
Convert major NPCs. This is where you get the largest differential, and you want your "bosses" to shine; it's not just hit points and attack bonuses but the cool new feats and classes and whatnot you want them to benefit from. I usually use Hero Lab to do NPC builds. (Many of the 3.5e Paizo APs have had fan conversions to Pathfinder posted on the Paizo forums.) My level 4 PCs can dogpile the heck out of a level 8 NPC boss if they're not done up well.
Sub in monsters where they exist in PF. Simple. Just look them up in the Bestiary(/2/3) and use those stats instead.
For lesser NPCs and monsters, either use tougher ones, more of them, or give them a quickie advanced template (or heck, just boost their hit points). For lesser NPCs, if you have the NPC Guide or Gamemastery Guide, there are a bunch of great NPC builds of various levels you can use for normal guards/bandits/soldiers/pirates/townsfolk/etc. and reskin as needed.
And that's it! I don't have the patience to do more, and unless you're publishing it there's no reason to.
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.
Best Answer
That is a pretty long list, but I will keep this focused on things in the Core Rulebook and Bestiary only: