Some interesting takes on the problems. thedungeondelver has some good insights, though I'm not sure I agree with his criticisms of 2e, though.
1e - I admit I'm not terribly familiar with 1st edition. But here's a start: The rules are sort of all over the place and some of the rules like the combat round, and initiative, and things of that nature are horribly confusing. A lot of the system required a lot of house rules to make coherent. Playing a cleric really sucked.
2e - 18/00 strength was weird. No strength could be over 25. They "dumbed" down demons and devils to appeal to Church mothers. It failed. Race/class restrictions, level limits. There were no actual grapple rules, and the wrestling chart meant that if you wanted to slug someone, you might end up performing a completely different result if you rolled on that chart. Very strange. Like most editions, 2nd edition had a massive problem with higher levels being way out of control, and wizards especially became unstoppable gods at the later levels.
3rd edition - the system is quite a bit more complex than what 2nd edition was. The power creep is still a problem, and wizards and sorcerers are gods at higher levels. Some of the same problems that have always existed still exist - "15 minute adventuring day", etc. The ruleset is also heavily codified and consistent, which a lot of feel tends to create a sort of "blandness" over everything.
4th - slaughtering cows is one thing, but 4th edition kills the whole damned ranch. The consistent rule "blandness" now blankets every class, and now every class looks and feels and plays almost like every other class, except the powers all have different names. Spells that have existed since 1st edition are now gone. Some "fluff" aspects are removed (Perform skills, for example), and the core rules removed certain classes (bards, barbarians, etc.) and added completely unnecessary races (dragonborn, eladrin).