It's impossible to know until we actually see the type of hockey this team is playing, and the growth/development of the young guys. Dan sounded really good that first summer too, we were hanging onto his every word. Same with Ted Nolan.
If we play possession hockey (I keep hearing up tempo and activation, but not support and only sometimes possession) then I think the improvement will be bigger than just the guys we added versus the guys we subtracted, and it will thus be bigger than what the general NHL media/fans who don't watch every Sabres game expect, as those people for the most part aren't even aware that the on ice system was a problem here.
As far as the question goes, it's difficult to name most improved; partly because of what Flagg says, and because teams that were bad had injuries or were simply not yet mature enough. Carolina will be a great team this year, but that's not just because of Williams and Darling; they were good last year too.
Calgary was already good, and now think Mike Smith is their savior, not sure I buy that
Arizona will improve but mostly because they had nowhere to go. Hjal helps a ton, but their goaltending, coaching, and leadership got a lot worse. Unless Domi, Duclair, and Strome have big years, they're not doing anything
Dallas would be my bet. I mistakenly threw Nill under the bus the other day, but he's had a good offseason. I don't like Radulov or Bishop for their contracts, but he should help them this year. That and the addition of Hitchcock will have the Stars much improved
I'm amazed Tampa isn't on this list honestly