E gives the biggest atk speed steroid in the game
AP is high burst and upfront dmg, but squishy. It's good for early/mid, but falls off hard late game when enemy AD can 3 shot you. You have to build lots of AP and almost no defense to keep up in dmg. After Rylai's Nerf, AP will be a harder sell than AD.
AD is much tankier and higher sustained damage late game, but much more expensive so takes more commitment. Trinity + Frozen Heart is a great build on Eve, grants 40% CDR, lots of mana for passive, and is great vs ADCs. A lot of tanky items are AD-based, plus you can build lifesteal to give even more survivability..
I really like AD eve a lot, and I build her AD more often than not.