It's been a while since I studied category theory so maybe my brain is failing me here, but unless I'm super-confused (1) products are unique up to isomorphism and (2) the product in Vect is the tensor product.
The cartesian product is the categorical product in some other categories you can see vector spaces as belonging to -- sets, groups -- but there is only one product in the category of vector spaces and it's the tensor product.
(2) is incorrect in that the tensor product is not the category theoretical product in Vect.
The product of Vect is given by forming the cartesian product of the vector spaces (with the usual vector space structure on it; and the usual projections into the components are the morphisms you need for the product).
The coproduct is the direct sum of vector spaces: In the product construction, take all those vectors where only finitely many coordinates are non-zero (but the morphisms you need for the coproduct are the embeddings of the components into the direct sum).
Of course, if you take products or coproducts of finitely many vector spaces, the two coincide.
The tensor product is neither a product nor a coproduct in Vect, as the universal property it satisfies is rather different from the product/coproduct ones.
(If you consider tensor products of algebras over a commutative base ring R, then the tensor product is the coproduct in the category of R-algebras, but this is probably not what you had in mind.)
> the tensor product is not the category theoretical product in Vect.
Oh, you're right and I'm a twit. (Not simply because I got it wrong, but because if I'd spent 30 seconds to think what the product construction does in Vect rather than relying on my plainly-unreliable memory, it would have been obvious that it isn't the tensor product: there are obviously no candidates for the required projection morphisms.)
Thanks! (And apologies to Jeremy for having made an invalid argument, though in fact I think the arguments "the product is the tensor product, the cartesian product is actually a coproduct, so they're very different" and "the product is the cartesian product, the tensor product is an entirely different kind of product, so they're very different" are about equally convincing modulo the fact that the first one's key premise turns out to be false.)
The cartesian product is the categorical product in some other categories you can see vector spaces as belonging to -- sets, groups -- but there is only one product in the category of vector spaces and it's the tensor product.
What am I missing?