Are there two non-diffeomorphic smooth manifolds with the same homology groups?

Sure -- there are an abundance of homology spheres in dimension 3 (the wikipedia article is pretty nice).

For other examples, in dimension 4 you can find smooth simply-connected closed manifolds whose second homology groups (the only interesting ones) are the same but which have different intersection pairings.

This last subject is very rich. For bathroom reading on it, I cannot recommend Scorpan's book "The Wild World of 4-Manifolds" highly enough.

A more trivial example is R^n and R^m for m and n different. (More generally two contractible spaces of different dimensions.)

More surprisingly, you can find smooth manifolds which are homeomorphic (and in particular, have the same homology) but are not diffeomorphic! The best-known examples are exotic spheres.