///The reason to call it B# instead of C is that it will be displayed differently on sheet music.

A note of B (equivalent to middle C) will be displayed as a B, not a C, therefore just above the A line in the Bass clef, and the # will be implied by the key signature (I hope).

If the key is C, the note will be displayed with a horizontal line through it.

But in reality it doesn't make much difference.///

It makes a big difference if you don't know there's such a thing as B-sharp.

If you're playing in A major where C is sharped from the key signature then anytime you run across a C, you play it sharp unless it has a natural next to it, then you play it natural. So if there's no such thing as a B-sharp and it's all C then every C that's a B-sharp has to have a natural sign next to it. That presents problems with notation in a key as A major such as when a C-natural is played, every C in that measure is natural unless notated otherwise. If want to C to return to sharp in that measure, you have to put a sharp sign next to it. Whereas if you use a B-sharp for C-natural, you can get around all this confusing notation trying to differentiate between C-natural and C-sharp. The C note will remain sharp by notating C-natural as B-sharp. That's much easier to read. But not if you're taught there's no such thing as a B-sharp.