I see no reason for $C(S^m, S^m)$, $C(S^n, S^n)$ to be homeomorphic. Here is a proof that their identity components are not homeomorphic.
The identity component of $C(S^m, S^m)$ is usually denoted $SG_{n+1}$. The homotopy fiber $SF_n$ of the evaluation map $SG_{n+1}\to S^n$ that sends a map to the image of the basepoint is homotopy equivalent to a component in the $n$-fold loops space on $S^n$. So $\pi_k(SF_n)=\pi_{k+n}(S^n)$, and the homotopy sequence of the fibration shows that if $n$ is odd, then the evaluation map is a rational isomorphism, in particular, $\pi_k(SG_{n+1})$ is infinite if and only if $k=n$.
If memory serves me it is not hard to compute rational homotopy type of $C(S^m, S^m)$. The homotopy type is harder to understand, but look for works of Vagn Lundsgaard Hansen (search in mathscinet with sphere in the title field).
ANSWERS TO SOME COMMENTS:
Ady: $SG_{n+1}$ is not the set of pointed maps, it consists of maps homotopic to identity.
I did not claim it enough to show that identity components are not homeomorphic. My answer was intended to push you in the right direction not to research the whole matter for you.
Reid Barton: I think knowing that different components of the $\Omega^nS^n$ are homotopy equivalent does not NOT imply that different comonents of $C(S^n, S^n)$ are homotopy equivalent.
If memory serves me, components of $C(S^n, S^n)$ are NOT all homotopy equivalent as shown eg by Hansen in the papers I mentioned such as [Hansen, Vagn Lundsgaard, The homotopy problem for the components in the space of maps on the $n$-sphere. Quart. J. Math. Oxford Ser. (2) 25 (1974), 313--321].
ONE MORE REFERENCE: Lupton, Gregory; Smith, Samuel Bruce, Criteria for components of a function space to be homotopy equivalent. Math. Proc. Cambridge Philos. Soc. 145 (2008), no. 1, 95--106.
