I know and understand how a pro is a lot better than a student (in most cases), but how much of a difference can there be between a $4,000 and a $6,000 saxophone? Would it be a big difference to a professional?
It could be worth a whole lot more than a mere $2000. If you feel good, you play good. If it looks good, you feel good. If you think that silver plated keys on a gold plated body is way cooler than a regular brass body, then that's all it takes for you to feel better. Consequently, you play better.
Of cause at some point the actual sound that comes out of the bell is important. The way the keys works, the action of it is important. But at $4000 it's all dam pretty good anyway. From there on it's more or less a matter of what brand you prefer. Which metal plating/ colours. Stuff like that. (Feel free to add a $1000 or two on mouthpieces too)
You sure manage to pick the hard ones. But I think that rudjarl is on the right track.
Last night I listened to a saxophone concerto by Takashi Yoshimatsu performed as part of the Australian Young Performers Awards. Its name is "Cyber-bird". According to the announcer, the concerto exploits all the possibilities a saxophone offers. It definitely seemed a challenging piece.
To tackle something like this in public one needs to have complete confidence in one's instrument and in one's skill as a player. Presumably a suitably qualified performer would give a creditable performance with something like a YAS-275. (For those not liking Yamahas, substitute your own preferred brand). But I kept wondering whether the performer might not feel more at ease with a professional-level YAS-875EX.