No, it's still pool! We also have snooker, which you wouldn't see in a pub; a working men's club or a posh club, maybe. Billiards is played on a snooker table but different rules/balls.

I learned to play pool in a working men's club but, get this, women weren't allowed to play on the snooker table as we 'might rip the cloth'.

What with, my stiletto heels? /sarcasm.