Who is a better support? Annie or Janna?

They're both good in their own right, Janna is better against hard engage comps. while Annie is played as more of an initiator herself.

It depends on how you want to play. Honestly I think Annie is a little more favorable in the current meta since she does more damage, has more kill pressure in lane, but I played a fair amount of Janna S3 (beat a Kha'Zix 1v1 in showdown, so it's not like she doesn't deal damage), and I really loved her kit.

Annie has more damage, Janna has more utility. If your team comp is lacking damage, I may go for Annie, but if you need more utility and especially disengage, Janna is good too.

You only listed one support (Janna) so I'll go with Janna. Annie is a mid people took bottom.

I tried both but I prefer Janna. Annie seems more like a one trick pony.

Annie brings stun(potentially AoE), long range AA harass, and damage. Janna brings a lot more protection, shield, and her utility. Janna also has the niche of rushing censer with a good turret killer like Jinx. Janna's AD buff + the AS buff from censer on an ADC makes turrets cry. Janna is still the safer support and better pick in a lotta situations, but I feel if you know your ADC is amazing and you can coordinate well, then Annie is a stronger pick. I voted Janna though as I feel she's better overall.