There is a lot of confusion on this because the "Big B" refers to the engraving on the horn, which of course has nothing to do with how it plays or sounds. There were a couple of substantive changes during the run of Big B horns, so you cannot generalize about them. The earliest Big B was essentially the same as the series one Aristo. Same horn except for ribbon key guards and Big B engraving. Somewhere in the middle of the run, the bell flare got larger, but otherwise it was still the same body and probably nearly the same horn (I'm not so sure about this one). Then toward the last of the run (still Big B engraving), the bell flare was increased even more, the bell itself changed shape, and maybe there were some other changes. This is the '156' Big B. Around '49 or so, the Big B engraving was changed to a script style engraving. This is the "series III" (156) and, according to Gayle at vintagesax, it is exactly the same horn as the latest Big B, but with a different engraving.
I have two horns that bracket the Big B run, a series one and a 156. They are both great horns but definitely different. Here's my point:
The early Big B is a DIFFERENT horn than the latest Big B (156). So you can't refer to the Big B as though it is one specific horn.
To answer the OP's question, the 1950 Aristocrat with script engraving is a 156, the same horn as the Big B 156 (the later Big Bs), but somewhat different than the earliest Big B with the smaller bell flare. These are all great horns. I think the 156 (Big B and script-engraved "series III") have a bigger, more spread sound, and are more 'free-blowing' than the series one and early Big B, which have a more focused, centered sound. And yeah, those are very subjective terms. I can't really decide which I like more.
Bottom line: You can't go wrong with a 1950 '156' Aristocrat tenor. It's the same horn as the later Big B, but different from the earlier Big B. Prices are almost meaningless. ALL Bueschers of this vintage are way undervalued, so you'll be getting more than your money's worth as a buyer!
p.s. When I had my 156 tenor in to the tech for a couple of new pads he told me the horn was so powerful he was afraid he'd blow the windows out of the shop! Of course he was exaggerating, but his point is right on.