According to the Stanford Encyclopaedia of Philosophy "Functionalism is the doctrine that what makes something a mental state of a particular type does not depend on its internal constitution, but rather on the way it functions, or the role it plays, in the system of which it is a part". If one adds the word 'functional' in this sentence, before 'type' it is hard to criticize, other than on the grounds that it is tautological and unenlightening. However, if by 'type' one either means what the state is like subjectively or how one might describe it in physical dynamic terms the claim must be wrong. This is straightforward for the physical dynamic terms, since these are presumably what are meant by internal constitution. For the subjective side the argument is a little more subtle.
Functionalism became popular because of arguments about things like octopus pain. It is claimed that an octopus very likely feels pain, since it shows withdrawal behaviour, despite the fact that its nervous system is significantly different from ours. It is therefore argued that 'pain' is multiply realizable in many sorts of biological system. However, we have no reason to think that octopus pain would feel like our pain so the argument seems empty. Pain as defined functionally can be attributed to many system with the same overall function, but that is the tautology. We have no evidence for a subjective experience being multiply realizable.
The key problem with functionalism, as I see it, is the concept of 'role in the world'. A role in the world for some entity X is an input-output relation. Output implies the influence of X on the world. If a mental state for X is something X experiences then we have a problem because X cannot experience its effect on the world, only the world can do that. X may get feedback indicating what its effect on the world may have been but that must be part of a subsequent experience. The content of a mental state, in terms of what is being experienced, can only depend on the way the world affects X. (See also Consciousness as Input.) Functionalism is causally incoherent.
The other problem with a 'role in the world' account is that it assumes that we can treat the world as being a constant, which is clearly nonsense. At any antecedent or subsequent point a 'role' can be bilked by changing a step in the causal chain. In the McGurk effect a soundtrack of a spoken 'b' combined with a film of someone speaking 'g' is heard as 'd'. In what sense is the content of the mental state of hearing 'd' determined by the role of hearing 'd' if it was 'b'? Similarly if the mental state belongs to the person filmed speaking 'g' how is the nature of this state determined by a role of being heard as 'd'? Put this way it seems quite ridiculous, presumably because our idea of an experience just isn't a 'role in the world'.
The simple fact is that no scientist would make use of this sort of definition of a state because it is completely useless for giving an account from which one can make reliable predictions. Science assumes that antecedent and subsequent events can vary however much one likes, so it seeks to find accounts that will work in all contexts - which are, effectively, accounts of the 'internal constitution' of dynamic events. Knowing that a screwdriver drives screws will tell you nothing about whether or not it will open a paint tin, but knowing what its internal constitution is will.