Would there be a difference between (aa)* and (a*a*)?
Is there a distributive property?
Kleene star does not distribute. (ab)* is very different from (a*b*).
In your specific example, (aa)* would match groups of two as (thus, it only matches even numbers of as), while (a*a*) is equivalent to (a*) and matches any sequence of as. (In that case, L((aa)*) is a proper subset of L((a*a*)), but this is not necessarily true for a general regex).