Sketches, using ideal hash functions
The obvious problem is that if you use a cryptographic pseudorandom number generator (PRNG), the correctness of your algorithm is conditional on a complexity conjecture. However, usually this can be avoided, because the full strength of cryptographic pseudorandmness is usually a huge overkill for streaming. If your streaming algorithm uses a small amount of space, then what you need from your PRNG is that its output cannot be distinguished from a stream of random bits by a small space algorithm. Such a PRNG was constructed by Nisan, and the proof of pseudorandomness is unconditional. 

A seminal paper in streaming algorithms that used Nisan's generator is Indyk's work on sketches of $L_1$. Since the sketch uses Cauchy random variables for which the mean and the variance do not exist, the usual moment calculations do not work, so it is hard to imagine how to use $k$-wise independence. That's why the heavy hammer of Nisan's generator comes handy.

A problem with these heavy hammers is that they come at a cost in space. The seed needs to be stored in memory, and for Nisan's generator the seed length is roughly $S\log R$ where $R$ is the number of random bits and $S$ is the space complexity of the algorithm. So you end up multiplying the space complexity by roughly the logarithm of the stream length $n$. If you can use $k$-wise independence you usually add a term like $O(k\log n)$ to the space complexity.