In "Quantum Computing verstehen", the author uses both $|ab>$ and $|a,b>$ to describe the state of a 2-bit quantum register. Is there any important difference between those two notations or do they basically mean the same?
It's the same in quantum information theory. They both mean $|a\rangle\otimes|b\rangle$. The second is just a little more explicit than the first since you're separating the two vectors by a comma. In quantum information this isn't so much a problem since $a$ and $b$ only take the values of $0$ or $1$, but in more general contexts (i.e. number states), the comma is imperative because it is difficult to discern the exact meaning of $|ab\rangle$. Take for instance $|231\rangle$ in the former notation in a composite number state system (like two harmonic oscillators combined). How would you interpret this? $|2\rangle\otimes|31\rangle$ or $|23\rangle\otimes|1\rangle$? With a comma in place it becomes clear what is meant and there is no ambiguity.
According to this article Composite bras and kets there is no difference.