Ok so first thing, RSA is a signature algorithm where DSS is a signature standard that currently uses the DSA algorithm with fixed parameters. For more details, see this great explanation.
So what we are actually comparing is RSA and DSA. The main difference is, that they rely on completely different hardness assumptions. I will try to explain them in a simplified manner as the underlying math is very complex, so please forgive me some abstractions.
DSA (as ElGamal) relies on the assumption that the discrete logarithm is hard in some special (sub)groups, meaning in those groups dlog is believed to be computationally hard.
In contrast, RSA bases its security on the (assumed) hardness of factoring, meaning given only a large number N that is the product of 2 primes p and q, it is hard to compute p and q from N.
So in the unlikely case that researchers
find an effective factoring algorithm for example, RSA is broken but DSA is still fine. In contrast, if dlog is easy, DSA can not be used anymore and RSA is still secure.
Given that, it seems to be a good idea to have several algorithms which rely on different assumptions just for the case that one is not true.
I hope this gives you a rough idea of the conceptual difference of those two signature algorithms.