Given that I just dropped an F-Bomb...

On my chemistry final,
would someone be so kind as to explain
to me the differences between the two branches
of mathematics known as Pre-Calculus and Calculus, respectively?

Pre-calc comes before Calculus, hence the name.  Seriously, it's just all the math stuff you need to be good at before you get to derivatives and integrals.  Algebra, rational expressions, number concepts (integers, irrationals, reals).