I think the comparisons in the OP aren't quite fair, as one's a live performance, while the other is a produced / mixed studio recording.
I'd say the better comparison would be Let it Go vs.
A filmed live performance lacks both, the polish of a studio record, but also the energy of hearing the actual singers with your own ears.
To me, DG feels a lot more empowering, and i'm also very biased towards songs that aren't entirely solo performances. But LIG is also pretty amazing as well.
Defying Gravity has always sounded a bit too self-consciously Broadway to me. When I hear it, I can just about picture Menzel looking out at the audience for a reaction.
Let It Go to me feels a bit more naturalistic, if that isn't too ridiculous a statement.
we can't really know, until we've seen Frozen - The Broadway Musical.
we do have
And, it's safe to say, that Idina's got a very similar ... intensity ... to her performance.