The main different between aniracetam and piracetam is the metabolization rate. Aniracetam is argued to be much more effective and fat soluble but has a shorter life span in the body. Piracetam is known to be less effective but is water soluble and has a longer - about 4 hours - effective time.

Are there any studies showing that this improves cognitive function in normal people?

It improves memory, not 'cognitive function'.

The category of "cognitive function" is generally pretty broad, and memory is generally accepted to fall under it. (I'm a neuroscientist, but if you don't trust me, do a Google search.) Memory without some kind of qualifier is not a particularly term either, since there are at least three forms of memory with distinct neural substrates.

FWIW, the closest thing I can find to a human study is http://www.sciencedirect.com/science/article/pii/00913057859..., which might indicate attention and/or working memory enhancement in non-human primates, but I don't have full text access. There are also a bunch of studies showing it enhances long-term memory in rats, but rats are not primates.

So why would I consider this over piracetam?