Can anyone give me a good example of two interestingly different ordinary cohomology theories?

For any space that has the homotopy type of a CW complex, its cohomology is determined purely formally by the Eilenberg-Steenrod axioms, so a counterexample is necessarily some reasonably nasty space. Here's an example you can see with your bare hands: consider the space $X={1,1/2,1/3,1/4,...,0}$. Now 0th singular cohomology is exactly the group of $mathbb{Z}$-values functions on your space which are constant on path-components, so $H^0(X)=X^mathbb{Z}$ (an uncountable group) naturally for singular cohomology. On the other hand, 0th Cech cohomology computes global sections of the constant $mathbb{Z}$ sheaf, i.e. locally constant $mathbb{Z}$-valued functions on your space. These must be constant in a neighborhood of 0, so the Cech cohomology $H^0(X)$ is actually free of countable rank, generated (for example) by the functions $f_n$ that are $1$ on $1/n$, $-1$ on $1/(n+1)$, and $0$ elsewhere, plus the constant function $1$.

I should add that topologists don't actually care about such examples. The point of the Eilenberg-Steenrod axioms is to show that cohomology of reasonable spaces is determined by purely formal properties, and these formal properties are actually much more useful than any specific definition you could give (the only point of a definition is to show that the formal properties are consistent!). What is of interest is when you remove the dimension axiom to get "extraordinary" cohomology theories, which Oscar talks about in his answer.