How do you prove that $\ell_p$ is not isomorphic to $\ell_q$?

To expand on David's comment, since the question he links to is not the same one that you ask: a theorem of Pitt (which is mentioned in the question Linked to) tells us that when $1\leq p < q <\infty$, every bounded linear map from $\ell^q$ to $\ell^p$ is compact. In particular, since the only Banach spaces with compact unit ball are finite-dimensional, there can be no bounded linear bijection between the two spaces.

Showing that naturally occurring Banach spaces are non-isomorphic can be surprisingly difficult; I don't know a simpler approach in this case, although one could probably rig up a direct argument by using ingredients from the proof of Pitt's theorem.

More information and some other non_-isomorphism results can be found in a MathOverflow answer of Bill Johnson.