There have been five stand-alone† editions of Savage Worlds:

1. Savage Worlds was printed in 2003 and was the first edition.
2. Savage Worlds (Revised Second Edition) was printed in 2004.
3. Savage Worlds Explorer's Edition was printed in 2007 and brought a significant rules revision. This also brought a smaller form factor and a low price, with the intention of making it an easy sell to curious GMs and their doubting players.
4. Savage Worlds Deluxe was launched in 2011. It is an expansion on the rules in SWEX and contains minor rules revisions, but is intended to remain compatible with (and more importantly, not invalidate) existing material or SWEX. The most important changes relate to rules for injuries and incapacitation, damage from area of effect weapons and a change to the Shaken mechanic. This is also higher page-count hardcover, so it's more expensive.
5. Savage Worlds Deluxe Explorer's Edition was launched in 2012 and is just the previous edition in a smaller, easier to carry around and cheaper format (same format as SWEX). However, it does have the errata from the larger Deluxe version included.

So the most up-to-date version is Savage Worlds Deluxe, but this is more a new version of the SWEX edition of the rules than a new edition. In the same way, the original Savage Worlds and Savage Worlds (Revised Second Edition) were mostly rules-compatible, so there have really only been two major editions of the rules, regardless of the number of books they've been contained by.

If you're using SWEX, you're using the most recent edition. This is the edition that all of the settings (first-party and third-party) and the Companions line of books are based on, and the Deluxe book this year indicates they're invested in keeping this rules base stable for players and third-party publishers for a while yet.

_{† Savage Worlds was evolved out of the rules for Deadlands and the Savage World of Solomon Kane, so it's arguable that there have been a couple of pre-first-edition editions of the rules as well.}

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.