Major Differences

1. the list of classes
2. the presumption of Non-Weapon Proficiencies
3. Advancement of Thief Skills
4. nature of Bards
5. Kits
6. Specialist Mages
7. Clerics
8. THAC0
9. Psionics

The list of Classes

AD&D 1E Core: Assassin, Cleric, Druid, Fighter, Illusionist, Paladin, Ranger, Thief, Wizard. Bard is special, see below.

AD&D 1E+ UA: Assassin, Barbarian, Cavalier, Cleric, Druid, Fighter, Illusionist, Paladin, Ranger, Thief, Thief-Acrobat, Wizard. Bard is special, see below.

AD&D 2E: Bard, Cleric, Druid, Fighter, Magic User, Paladin, Ranger, Specialist Mage, Thief. Barbarian and Cavalier still exist, but as kits, which see below.

Non-Weapon Proficiencies

While NWP's exist in late AD&D 1E, they are presumed to be optional add-ons, and not listed in adventures.

In AD&D 2E, while technically optional, almost all examples and almost all pregen characters include the Non-Weapon proficiencies. They are presumed as a part of the game line design. This is a huge change in the nature of adventures, too. The use of NWP's is expected in some adventures, and explicitly required for a few more.

Thief Skills

In AD&D 1E, thief skills advance along specific tracks, and all characters of a given level have the same ones have the same base, modified for race, armor, and attributes. This also means NPC thieves do not need their scores listed, as they can be figured from the DM Screen on the fly.

In AD&D 2E, thief skills have a base at 1st level, but a pool of points added to that base at 1st level, and a smaller pool at each level thereafter. Thief skills must be listed for NPC's, as it's much harder to assign on the fly.

Further, in later 2E materials (Dark Sun, Skills & Powers), there are additional thief skills added, and PC thieves pick which ones they take at first level, and gain the remainder at 9th.

This also affects Bards, as in 2E, bards gain certain thief skills for being bards, and use the same points per level method as thieves.

Bards

In AD&D 1E, the Bard in the PH can only be taken by dual-classed fighter/thief characters. The Character must be between 5th and 8th level as a fighter, then 4th and 7th as a thief, and then dual class into Bard. This requires some insane stats, and extensive play. Bards will likewise have extensive thief abilities, be competent fighters, and will not gain more HP for several levels due to the dual classing rules.

In AD&D 2E, Bards are a core class. The thief skills are a subset, not the full range. Fighting ability is weaker than fighters. HP are comparable to thieves.

Historical Note: The original Bard class in Strategic Review was closer to the 2E bard than the 1E presentation, but the details of ability were comparable to using the 1E bard as a core class.

Kits (2E only)

The concept of Kits is mentioned in the 2E Core Rules, but they are not presented until the Player's Handbook Rules Expansions (PHBR series). A kit has a set of requirements, provides some bonus proficiencies, and occasionally, bonus special abilities. Many were somewhat extreme.

The equivalent role in 1E was filled by specialized subclasses presented in magazines, as exemplified by the Cavalier...

Specialist Mages

AD&D 1E has 4 specialist mage classes, with only one, Illusionist, in the core rules. (The other 3 are in the Forgotten Realms Adventures rulebook.) Illusionist is presented as a full-up core class; the Forgotten Realms ones are full from the 3rd level on, and require core Magic User for levels 1-3. A few additional specialist classes appear in magazine articles.

AD&D 2E presents 8 specialist wizard subclasses as a single core class in the PHB. They differ from each other only in specific spells and attribute requirements.

There is no specific specialist spell lists, but every spell has specific school attributions, and those schools are the basis for the specialist classes. All specialists spells are available to core magic users.

An additional variant class is presented in Tome of Magic, the Wild Mage.

Clerics, Priests, and Druids.

In both games, both cleric and druid are presented as a core class.

In AD&D 1E, they have separate, and only somewhat overlapping, spell lists.

In 2E, both use Priest Spells. 2E Clerical spells are assigned to specific Spheres, with a number being in more than one sphere. Clerics have several spheres; druids have a specified set of spheres. Provision is made in the PHB to allow for creation of similarly specialized priest classes; further details are in the PHBR for Clerics...

Since the whole of the priestly spell list is unavailable to core priests, this makes it more difficult to select what spells are available to a given priest... but it's also now a smaller list for any given priest.

THAC0

THAC0, "To Hit Armor Class 0," was a shortcut used in some later AD&D 1E materials, which imperfectly reflected the AD&D 1E To Hit tables, with their flat spots.

THAC0 was adopted as the official mechanic for AD&D 2E, and the To-Hit tables reworked to make use of it.

This may seem trivial, but it makes negative AC's much harder to hit for low level characters, as AD&D 1E has a 6 AC entries for a To-Hit of 20. So a Modified roll of 21 in the tables hits an AC 5 lower than using THAC0. (As a reminder, the modifiers were situational and attribute only; THAC0 itself is modified in AD&D 2E, and table of roll needed is indexed by Level (on a class basis) and AC being attacked.

Psionics

In 1E, psionics are in the core rules, in an appendix, relatively unchanged from their Original Edition version in Eldritch Wizardy (Supplement 3).

In 2E, psionics are in a PHBR rulebook, not the core rules. The mechanics get reworked entirely, and while having throwbacks to the older rules, they are quite different in execution. The use of proficiency-score checks, and the methods of generating Psionic Points are very different.

Lesser changes

• The specific modifiers for attack rolls have changed.
• Many specific spells have significant explicit changes
• specific wordings changing resulting in different interpretations on many spells
• specifics of the Weapons vs Armor tables differ within AD&D 2E; they are different as well from AD&D 1E.
• Specific entries for the XP earned by non-combat methods.
• Many monsters have changes, sometimes extensive and substantial. Especially Dragons.
• Angels, Demons and Devils are not called that in AD&D 2E. Baatezu and Tan'ari are demons and devils.

So what's the same???

Mode of play remains unchanged. The relationship of the Initial 3 classes (Fighter, Mage, Cleric) remain the same, and the Thief as well in relation to those. The Druid, Paladin, and Ranger as well retain their core character.

The general modes of advancement are the same, even tho the specific methods of earning XP have been expanded, and the XP tables are close (tho not always identical).

The basic mode for magic is still the same, and is still spells per day.

The save categories remain unchanged. The unique monsters - illithids, rust monsters, beholders, and several others are the same as ever in general terms, even if some specifics vary.

Bottom Line

2E is a different game from 1E, but shares much of the heritage. They're able to borrow across, but rules as written, they are not the same games. There was far less difference between Original Edition D&D as expanded and AD&D 1e than between 1E and 2E.

Somewhat oddly, your question is answered by the Sage Advice Compendium document which was first released as part of a Sage Advice article by Jeremy Crawford on the official D&D website entitled Rules Roundup.

This document says the following on the first page, under the "Errata" heading:

A corrected version of the book includes the following text toward the bottom of its credits page:

This printing includes corrections to the first printing.

At the time of writing this answer, there have only been two printings of the Player's Handbook: the original, and the latest which incorporates the errata updates. Any rules clarifications which are not considered errata are going to continue to be found in the updated versions of the Sage Advice Compendium document. The latest version is always accessible from the official D&D website, and includes a version number both in the file name and in the footer of the document. The latest version always takes precedence over previous versions. Note that the errata document can also be distinguished from later updates by its version number, which is also contained in the file name as well as the footer of the document.