You got her w and e messed up and I'll explain the power of her abilities.
Her q is a glorified auto attack at best, about half as strong as a single one from an adc but that is only vs minions as it has a longer cooldown when not used to cs.
Her w is a VERY short ranged aoe damage ability that can only be used vs stunned targets, minions, or certain melee champions (many melee champs have longer ranged abilities than this). This ability adds to her burst but can't really initiate it while staying alive.
Aaaaand her e. Probably one of the most useless abilities in the game. Only purpose is to provide her passive. Does not deal damage unless someone bursts her when it's up and even then it will not deal over maybe 100 or so. It will not save her because she is squishy. It does not add to her burst. It doesn't even speed up tibbers anymore which used to be a somewhat ok excuse to have it in the game. You comparing it to thorn mail would be crap even if it was that good, but it doesn't give as much resist, deals almost no damage, and comes prebuilt on a squishy.
I hope you realize now that Annie has three abilities, two of which are decent at best, and she has tibbers. That is all.