At the moment, my piston Bb is a Monette Prana3 and my C is a Bach 229H, so EVERYTHING is different.
If we take 2 "standard" Bachs (a model 37 and a 229H) and compare them:
the C has a more fluid "creamier" sound. The Bb is more brazen, with a different type of edge when pushed. The C sound is perhaps "clearer". In any case, what happens to the overtones when playing a crescendo is much different. The Bb gets a much different type of edge - starting at lower volume levels. The strings start to complain sooner with the Bb as the edge means more overtones and that masks what they hear.