After I pause should shots still be hitting home?
Oh, bugger, I think that's an error. I'll check on that and fix it, but no, they should not be doing that.
What's the difference between TBS pause and the other kind?
TBS pause basically keeps running the internal "turns" behind the scenes, and takes player commands and such, just preventing ships (and should be shots also) from moving, firing, etc. This makes it so that you can place ships, adjust build queues, place flares, set attack targets, and do a lot of other things like that. I'll add this to the wiki with more detail tomorrow, but basically this mode is probably what almost everyone would want to use for pausing (especially in multiplayer, when it comes to flares), except when you are wanting to send a desync report if one is needed...