It's easy to see that $\operatorname{Hom}\left(\bigoplus_i M_i, N\right) = \prod_i \operatorname{Hom}(M_i, N)$. However, there are a couple of ways this can conceivably fail if we replace the coproduct on the left with a product: we could have a homomorphism $\bigoplus_i M_i \rightarrow N$ which didn't extend to $\prod_i M_i$, or we could have one that extended non-uniquely.
In the latter case, there would be a nonzero homomorphism $\prod_i M_i \rightarrow N$ which was identically zero on elements of "finite support." Can this happen, and if so is there a nice example? (Apologies if this is obvious -- I've been thinking about it for a while and can't come up with anything.)