Hmmm....
Didn't play first edition (which you are clearly referencing, even though 2e is ALSO AD&D) so I will intro my "Top Ten Reason 2nd edition is more-better"
1.) The Settings. Oh Gawd, the settings. In NO other edition of D&D could you chose between Greyhawk, Realms, Dragonlance, Mystara, Ravenloft, Planescape, Dark Sun, Birthright, Spelljammer, a dozen historical settings, and a couple of micro-settings (Jakandor, etc).
2.) Organized for your use. Need to find the encumbrance rules? Just need to divide up XP? No need to read a full page diatribe on what encumbrance means to medieval peasants (complete with
World of Greyhawk plug), they are easy to find.
3.) Non-Weapon Proficiencies: Granted, not the best skill system ever created, but much better than "let your DM decide if you can swim".
4.) Less nonsensical rule-contradictions. Ok, So Strength adds to my hit and damage UNLESS I'm a monk, which requires me to have 15 strength YET I get no bonus for it? Ok. I'll be a half-elven bard. Half-elves can be bards ya know, despite the fact they can't dual-class to BECOME a bard. Oh, and Why do I need to take 10+ levels of other classes to BE a bard? Oh, and why aren't GIANTS considered GIANT-CLASS monsters? And why does the cavalier get to bump HIS ability scores by level?
5.) The Best Monster Manual D&D has ever put out. Period.
6.) More supplements than you'll ever need. Concerned about the lack of new 1e/3e supplements? Fear not, TSR put out so many from 1989-2000, people STILL haven't used them all. Still dying? Ok, OSRIC/1e is easily compatible with 2e as well.
7.) Genre support. Gritty historical. Check. Epic fantasy? Check. Sword-n-sorcery? Check.
8.) Some of the best D&D video games (Baldur's Gate, Icewind Dale) utilize 2e rules.
9.) AD&D Core Rules 2.0 + Expansion. All the features of DDi Char-gen, plus custom content, mapping, and 20 books in HTML.
10.) Specialty Priest and Spheres: Never has customizing priests to fit their god's ethos been done so well.