From what I understand, Thor was the God of Thunder before he was given Mjolnir. Were the powers of controlling thunder and lightning transformed into Mjolnir, so that now he no longer has them (since he doesn't wield it any more)? Or were the powers duplicated when the Hammer was created, so that now both the old and new Thor can control Thunder and lightning?
In any case, in a more abstract level, the new Thor is a human. Is she really the Goddess of Thunder, or is that just a nickname, like Thor is just a name the old Thor decided to give her?
I don't think this is a duplicate to this question, because the fact Thor could control thunder and lightning before he was given Mjolnir doesn't mean he still can after Mjolnir left him. It does say whoever holds the hammer "possesses the power of Thor", but this sentence is a bit ambiguous; I mean, if whoever holds it is called Thor then it's pretty much meaningless. Perhaps 'Thor' in this sentence refers to the original Thor, but that would mean Mjolnir only grants the powers Thor possesses anyway, and he definitely became weaker after loosing it.