I have been reading in Fulton & Harris's book on representation theory and it talks about things like the decomposition of a direct product of representations $ V \otimes V $ into a direct sum of representations. It seems to me there is a real difference between a finite direct sum of representations and a direct product even though they agree on vector spaces. Is this difference real and if so , what is it?
$\otimes$ denotes the tensor product, not the direct product. This is different even for vector spaces; the tensor product of vector spaces of dimensions $m, n$ has dimension $mn$ rather than $m+n$ for the direct product or direct sum.