PasswordDeriveBytes implements the PBKDF1 key-derivation function. A KDF is a function which transform a piece of secret data (here, a "password", i.e. the kind of data which fits in a human brain and can be typed with human fingers) into a sequence of bits adequate for algorithms which need a symmetric key (e.g. symmetric encryption). A KDF is not meant for anything else, in particular password storage.
A hash function can be used as a KDF, provided that the symmetric key you need is no longer than the hash function output size. However, such a KDF would be very crude. One feature which good KDF should provide is to be adequately slow. This is because "passwords" are, by nature, vulnerable to exhaustive search (also known as "dictionary attack": users tend to choose as passwords rather simple words or combinations which can be guessed with only a few millions or billions tries). In a given system, one can usually tolerate a relatively slow KDF: a user trying to authenticate will not see the difference between a 1µs and a 1ms delay for the key-derivation function; but a 1000x slowdown is deadly for the attacker: it converts a one-day breaking effort into a three-years breaking effort.
PBKDF1 includes an "iteration count" which is a parameter meant exactly for that: to make the key derivation adequately slow, in a configurable way. A simple hash function is just too fast for that. Usage as a KDF is precisely where you would prefer PBKDF1 over a hash function. Actually, PBKDF1 is not recommended; PBKDF2, from the same standard, is supposed to be more robust.
Hash functions are much more generic objects than KDF, and have many other usages, which KDF do not fulfill.
What you want to do is unclear: you use the term "signature", which normally means "asymmetric digital signature" as with RSA or ECDSA; there are some people who tend to use the term "signature" to designate a symmetric integrity check, such as a MAC (calling it a "signature" is improper, but widespread). However, this entails some piece of secret at some point, a key, and a hash function is key-less.