PRF and pairwise independent hash function
You are right; these are the properties of a PRF. In fact, a $k$-wise independent hash function has exactly the same distribution as a truly random function, as long as you only see up to $k$ points. This is the difference: a pseudorandom function has to be indistinguishable from random for any polynomial number of samples viewed. This is both weaker and stronger than $k$-wise independence: a PRF is only computationally indistinguishable from random (weaker), but maintains this property for any polynomial number of samples.

Note also that $k$-wise independent hash functions become very inefficient when you look at large $k$. So typically they are very useful when you only need pairwise independence or a small $k$.