What are the differences between the oldest (EON) version of Cosmic Encounter and the newest (Fantasy Flight)?
Here are most of the major changes (including expansions).  (Since a lot of Eon Cosmic fans also played the Mayfair 2nd ed, I've mentioned a it where relevant.)

Main gameplay


The basic game mechanics are unchanged.
Hand size is now 8.
Flares are Eon-style, not Mayfair-style (once played they return to the hand, and are only lost when a new hand is redrawn).
FFG counters allows for up to 8 players with all expansions.  Team play rules have just been added for once you go over 6.
The fixed player boards of Eon/Mayfair are replaced by separate coaster-sized planets.  This seems like a cosmetic change but it's not; several powers and flares now take advantage of the fact that planets can be moved around independently.  (e.g. one alien power actually steals whole planets.)
The main deck contains positive attack cards, negotiates, reinforcements, artifacts and flares.  The confusing array of Eon / Mayfair zaps have been simplified down to the Cosmic Zap (unchanged), and the Card Zap, which zaps artifacts, flares, and zaps.  There's also a 'morph' challenge card, which duplicates your opponent's challenge card.


Optional rules


Destiny cards:


In addition to the wild cards, three special destiny cards are included which all pick a 'well-off' player (e.g. fewest tokens in the warp).
The 'rewards reversed' destiny cards have been improved into an entire expansion mechanic, the hazard deck.  One card in each colour is a 'hazard' card which calls for an alternate rule for the encounter to be drawn.  This deck includes the old 'reversed ally rewards' rule, but also many others that change the encounter feel and require tactical adjustments (e.g. one hazard sends both players to the warp if the total of their attacks is too high).  Three of the cards are long-lasting; they change the rules for all encounters until another one of the three replaces them.

Removed: Most of the more random or bookkeeping-intense optional rules.


In particular, moons and lucre are gone. (Along with all powers that needed them.)

Added: Several major optional card decks.  The hazard deck is mentioned above; there are also:


Tech research.  If this rule is used, each player starts with a random tech card, which can be researched by putting tokens on it at the rate of one a turn until its cost is paid.  This means having fewer forces available on-board, in return for a reward later.
These powers can be low-cost and small, or expensive but extremely powerful.  (e.g. one card creates an entire new planet.)
Rewards deck.  Intended to make being a defensive ally a stronger option; defending allies may draw rewards from the main deck or this one.  This deck includes attack cards (including some negative cards, and not going as high as the main deck attack-30 and -40, but averaging pretty high), crooked negotiates which penalise your opponent more for not dealing, kickers, artifacts, and some cards which retrieve lost ships.  Those cards in this deck also explode and destroy tokens if an opponent ever steals them from you.  (Reward cards have different backs, to discourage being taken by theft powers.)



Powers


In general, the full range of classic powers have been included, but where there were several similar powers, FFG have picked only one.
Powers are rated by complexity, so that the hardest-to-use can easily be removed for beginners' games.
There are no powers that depend on certain optional rules being in play.  Any power will work in any game.  Some powers get better with certain rules in play, of course.
Powers or flares which had ambiguous or interpretative effects (Judge, Witch, etc.) have been reworded to be unambiguous.  (For old-school Eon players, some of the reworded flares have the original version included as an alternative.)
A significant number of new powers are added.  Generally they're well-designed, and surprisingly original - several do odd things with the basic game mechanics, like destroy planets or change the victory conditions.  Many are previewed in the news feed at Fantasy Flight's Cosmic Encounter website, so check there for examples.