It's easy to construct myriads of such examples using finite $T_0$ topological spaces. These are equivalent in a natural way to finite posets (check this wikipedia article). With a poset $P$ we may associate an abstract simplicial complex $K(P)$ whose simplices are the chains of $P$. It turns out that the geometric realization of $K(P)$ and $P$ are weakly homotopy equivalent (there is a continuous map $|K(P)| \to P$ inducing isomorphisms on homotopy groups). However, these spaces are not homotopy equivalent (unless they are both homotopy equivalent to a discrete space).
Moreover, examples of weakly homotopy equivalent finite spaces that are not homotopy equivalent are also easy to give.
The following notes by J.P. May are a nice introduction to this topic: http://www.math.uchicago.edu/~may/MISC/FiniteSpaces.pdf , http://www.math.uchicago.edu/~may/MISC/SimpCxes.pdf
