another clever way to show the identity map of a 2 - sphere say, is at least not homotopic to a constant by a differentiable homotopy, is to note that the solid angle form has ≠0 integral over the sphere. It is easy to show by Stokes theorem that two differentiably homotopic maps will pull back a closed form to have the same integral. Since a constant map pulls back any form to have integral zero, this would do it too.
These two ideas, homotopy and integrals of closed forms, become the basis for a whole theory of studying spaces, called homotopy and homology. There is a way to actually "multiply" two maps from an n- sphere into the same space, and make the set of homotopy classes of maps into a group, the nth homotopy group. It is even easier to make the space of closed n forms into a vector space. When modded out by the exact forms one gets a space called the nth (deRham) cohomology space.Then a really short answer to the question above is this: S^n and S^m cannot be homeomorphic for n < m, because both the nth homotopy and cohomology groups of S^m are zero, whereas for S^n they are both non zero.You learn the homotopy theory in a first course in algebraic topology, but some people teach the cohomology theory in advanced calculus, e.g. Ted Shifrin's book, or Michael Spivak's Calculus on manifolds.
Notice the proofs above reduce down to the easy theorem that 0 ≠ 1. Using the more subtle fact that 1 ≠ -1, we can prove that a 2-sphere does not have any never zero vector fields, since that would make the identity map homotopic, not to a constant map, but to the antipodal map, which has degree -1.
The reason you can teach this stuff before algebraic topology and without doing homology theory first, is that for a sphere the cohomology group is generated by the solid angle (or volume) form, so you just write it down. On a 1 - sphere that's "dtheta".
I.e. the whole ball of wax boils down to showing the identity map of a sphere is not homotopic to a constant. If it were (in dimension one) you would get zero when you integrate dtheta around a circle, but you don't, you get 2pi. The analogous argument works in all dimensions. I.e. an equivalent statement is that there always exists on every sphere, a closed form that is not exact, and you can just write it down.
see e.g.
http://en.wikipedia.org/wiki/N-sphere for the "volume form".
the part about getting zero when you integrate an m form around the n sphere is even easier since the way differential forms work, pulling an m - form back to n - space gives a form which is identically zero when n < m!