Terrible early game is really the only reason.
It's been a few weeks but I LOVE AP Poppy. Unfortunately, wave clear is quite possibly the slowest wave clear in the entire game, due to this you often end up being forced to farm under turret and let the enemy laner roam. Ooocasionally you get that odd guy who doesn't even know Poppy is a champ and constantly tries to fight you next to walls and you can just bully him out of lane, but aside from that, playing Poppy is like admitting you lost lane before minions spawn while holding up a giant sign that says "Soon." in tiny font. ... At least that's what it's like with AP Poppy.
I don't play AD Poppy, sorry. If I wanted to play an AD melee hyper carry I'd play Jax.
As for why I haven't played Poppy much lately, I'm just waiting for her rework. I just hope they
A) Don't change Lollipoppy's splash art and
B) Keep AP Poppy possible.