Functionalism vs Formalism?

Perhaps a better way to approach the difference between functionalist and formalist approaches would be to explore the history and personalities involved.

In principle, there's no opposition between functionalism and formalism. Units of language are used for certain functions and these functions can be described using formal methods. The question really is what are those functions and what are the formalisms used.

Historically, functionalism was closely associated with the Prague structuralist school of linguistics (known as the Prague Circle of Linguistics - full disclosure, I'm a member). The functionalist theme developed in the 1930s (later put into in opposition to the functionalism of the Copenhagen school approach of Hjelmslev who understood function more in the mathematical sense). The Prague school was revived in the 1960s and tried to blend functionalist methods into the then increasingly popular formalist approach (inspired for instance by formal semantics of people like Montague).

Formalism, on the other hand, is associated with Chomsky (although there were others and even earlier schools) who developed mathematical theories of the combinatorics of grammar combined with the claim that they are facts of language without any reference to their function. But other formal approached to language were also being developed about the same time. I already mentioned Montague, but there was also Bar-Hillel with his categorial grammar. Both of these were much more compatible with functionalist approaches in that they are concerned with formalizing language units that are put to real uses.

Today, functionalism is most closely associated with the British Firthian school of linguistics - it's most famous proponent being MAK Haliday (it is now particularly popular in Australia and New Zealand). This school had the most impact on wide areas of language description from text and discourse analysis to pedagogic grammar (most foreign language textbooks and grammars will be beholden to some version of functionalism). Most of the work on language corpora has come out of the functionalist tradition (in the broadest sense). I think Haliday's three meta functions of language: 1. Ideational, 2. Interpersonal and 3. Textual are still the best delineation of what dimensions a linguistic theory needs to account for.

Ultimately, the difference between formalist and functionalist linguistics is not the irreconcilable rift between linguistic theories it is thought of as. I'd suggest any one linguistic theory needs to be judged on a range of issues and whether it is labelled or labels itself as functionalist or formalist, is probably not that important.