[mturmon]

Yes.

And beyond this intuition (escape from local optima), the reason that annealing matters is that you can show that (under conditions) with the right annealing schedule (it's rather slow, T ~ 1/log(Nepoch) iirc?) you will converge to the global optimum.

I'm not well-versed enough to recall the conditions, but it wouldn't surprise me if they are quite restrictive, and/or hard to implement (e.g., with no explicit annealing guidance to choose a specific temperature).