I used to play Axis & Allies all the time when I was younger. I recently purchased the Axis & Allies Revised Edition, and was floored by all the added complexity/new units/new rules/map changes. It was definitely a whole pile of fun, but I notice now that there are many other versions of Axis & Allies available as well. Are the "new" rules the same across all the "new" versions? Does each version have its own rules? Are the differences in the rules relatively minor, or do I need to keep track of six different sets of infantry rules for six different games?
I'm not sure the best way to answer this question, but I'm going to propose a running list starting with A&A 2nd edition as the baseline and then listing each progressive revision below it with all the changes from the previous version--just like software release notes.
Since I don't own all the versions I'm making this answer a community wiki. Please add to the list if you see something missing. I'll start us off with the easy stuff.
3rd edition changes
I own Axis and Allies 1984 original, the 2004 and 2009 revised editions, as well as the new Europe and Pacific games.
The new rules, new units, new everything make the game so much more interesting.
For example, in Europe and Pacific, there are "Harbors" and "Air Bases." These allow units that move from them to move one extra space. Let's say a fighter is in Hawaii. Fighters have a move of 4. If it leaves from an area with an "Air Base" however, its mobility is upped to 5. The same is true for bombers (and the new tactical bomber). These bases add one movement to any unit. Harbors allow all ships leaving the harbor to move at a 3.
If you want more info on these versions, please comment back.