It’s the correct answer, but not defensive enough for my liking. 🙂 So treat this answer as a restatement of his.
There are several ways of explaining why language is the way it is. Originally, the split was between diachronic and synchronic explanations. The diachronic account, which is historical linguistics, explains language in terms of earlier stages of the same language, and change processes. The synchronic account, which took over as the mainstream with Saussure, explains language as a system in its own right, rather than seeking to explain it in terms of process.
Since maybe the 70s in some quarters, but the 50s in others, there is a related split.
- The formalist account of language explains language as a system in itself, without appealing to extralinguistic causes. An explanation in formalism is the formulation of rules that explain the distribution of phonemes and words and phrases. Generative grammar is the major class of formalist accounts. It ultimately appeals to a language device in the brain: language is the way it is, because that’s how the rules for linguistic structure in the brain work.
- The functionalist account of language explains language as a means of communicating meaning. So giving the rules by itself is not enough in functionalism: functionalism want to know why those rules, and not others, are best suited to communication. The rules end up having a lot to do with pragmatics and semantics and discourse structure, as Trevor says; and ultimately functionalism concludes that language is the way it is because of cognitive patterns in general, and not a part of the brain specific to language. If you think about it, that also means functionalism is a lot friendlier to diachrony.
They’re incommensurate approaches; *shrug*. To a functionalist, formalist accounts don’t really explain anything, and are circular. To a formalist, functionalist accounts are specious Just-So speculation, and are unscientific.
(I’ll only disagree with Trevor in one detail: functionalists in my experience love typology—it gives them more things to explain in their terms.)
The home turf of functionalism is the West Coast of the US, and it was also big in Australia when I was going through the system. Systemic functional linguistics is an earlier branch of the theory, developed in the UK and Australia (though restricted to Sydney Uni there), and which other functionalists don’t like. It is very popular in applied linguistics, as it gives paedagogically satisfying accounts of language variety.