I know that there definitely are two topological spaces with the same homology groups, but which are not homeomorphic. For example, one could take $T^{2}$ and $S^{1}\vee S^{1}\vee S^{2}$ (or maybe $S^{1}\wedge S^{1}\wedge S^{2}$), which have the same homology groups but different fundamental groups. But are there any examples in the smooth category?
4 Answers
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 simplyconnected 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 4Manifolds" 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.)

$\begingroup$ I was thinking of a circle and an annulus, you beat me :) $\endgroup$ Jan 6, 2011 at 0:29
More surprisingly, you can find smooth manifolds which are homeomorphic (and in particular, have the same homology) but are not diffeomorphic! The bestknown examples are exotic spheres.

1$\begingroup$ I actually knew that. For some reason, I didn't make the connection between homeomorphism and same homology groups. Thanks! $\endgroup$ Oct 30, 2009 at 4:11
Serre has shown with the help of two embeddings phi and psi of a quadratic number field into C that there exist two projective surfaces V(phi)and V(psi) over C which have non isomorphic fundamental groups (and so are non homeomorphic) but have isomorphic Betti numbers.
The comparison theorem between étale cohomology and singular cohomology ( which didn't exist when Serre wrote his article ) even proves thar these surfaces have the same singular cohomology with value in any finite abelian group or over Q_l(ladics) for any prime l.
I don't know if these surfaces have the same homology and so I don't answer your question in the strict sense (anyway, now you have Andy's and Eric's most satisfying solutions); but these remarks in an algebraic geometry context might interest you. Serre's article is
Exemples de variétés projectives conjuguées non homéomorphes, C.R. Acad.Sci.Paris 258 (1964), 41944196
It is of course reproduced in his Collected Papers.