Question

Why did we have to say f is continuous on [a,b] and differentiable on (a,b) when we could just say f is differentiable on [a,b] (which implies f is continuous on [a,b])

Answer 1

I see that a lot in Calculus

Answer 2

and in Real Analysis

Answer 3

The first way says that its continuous on the whole interval but only differentiable on the open interval, so not at the end points. However, the second way "f is differentiable on [a,b] => f is continuous on [a,b]" is saying that its both differentiable and continuous on the whole interval.
As for why it matters that its differentiable only on the open interval, not the closed, I can't say.