Scharnhorst has slightly more range with the second set of trip 11"s mounted than the moltkes Kust dual 11"s. The scharnhorst is very big target though.

I'd suggest using the Moltke till AD. You can have the firepower of a BB1 without taking up a BB slot, letting you jump into games quicker (.i.e. rooms with 20 bb slots already filled) or you can use the Moltke for AA. If you dont have KM AA gunners, the regular dual 4.1 guns work pretty well for AA.

I definitely recommend using it for AA, you get a lot of exp, credits and experts for your crew for doing it, which I think is more useful than a BB1 taking up a slot that a potentially high-tier BB could have instead. Its a flexible ship design, so you can mount AA in the T slots + Dual 11s in the R slots, or HH instead of the Dual 11s for ASW, or even both, with Dual 11s on the front R slots, HH on the back R slots, and AA on the T slots.

This somewhat depends on your crew. If you have a great crew in the BB1, you may be able to pull some high attack games off, and get more XP. Otherwise, AA with the moltke is the way to go. (Personally, I hated the moltke due to it's lack of firepower. Then again, playing KM after IJN makes every ship seem weak.)

Definitely Moltke. It is a smaller target, with just one less gun, and has great AA capabilities. I hated the Scharn for a few reasons. First, although I could armor whore it to hell, I was slow as an Elefant tank. Lots of firepower, big target that even the other BB noobs who are only packing 10 vets per gunner can hit. Speaking of vets, even with 160 vets/350 experts on the Scharnhorst, spread was terrible on the triple 11" guns. Moltke has much better spread, sacrificing a little time and range for accuracy. A few more vets and I would have been block shotting. 
Moltke, no question. More AA, smaller hitbox, slimmer profile, and much faster. If there was a showdown between a level 120 Moltke crew and a level 120 Scharnhorst crew, I'd put my money on the Moltke every time.