I haven't got chance to play Zyra yet, but I've played Ahri since the day of her release so I can say a few words.
Ahri is much more mobile than zyra thanks to her ulti. She can easily win 1v1 against her by bursting her down with all spells. Zyra has to preapare much longer for fight, but when you can turn all of her skills very quickly( plant 2 seeds, get q on them and champion, get grasping roots at enemy champion and follow it all with ult) you have really big chance of winning.
They're both good at teamfights: massive aoe, good cc.
Zyra has much better cc and is a great help when enemies are close to each other. Ahri is a little more usefull when enemies are trying to escape from teamfight thanks to her low cooldowns and ult. She is better at bursting down enemies.
I prefer Ahri, but both are very good mages and safe picks for mid.
I hope I helped!