I’m not contradicting Warren M Tang (see Warren M Tang’s answer to How we can differentiate functionalists, cognitivists, and structuralists?), but let me try a different formulation.
- A functionalist explains language structures by appealing to the communicative function of those structures. (They do linguistics by metaphors.)
- A cognitivist explains language structures by appealing to general psychological processes of cognition. (They do linguistics by diagrams.)
- A structuralist explains language structures as a coherent system of signs. (They do linguistics by tables.)
These approaches are not mutually exclusive in principle—though they tend to be in execution.
Where’s Chomsky fit in all of this? He wishes he was a cognitivist; he’s actually a structuralist.