Q:

What is the difference between expressivism and representationalism in modern philosophy of language?

Philosophers like Robert Brandom and Huw Price make a fairly sharp distinction between expression and representation  (or at least expressivism and representationalism). Price goes so far as to recommend a wholesale rejection of representationalism in favor of expressivism; Brandom tries to rescue representionalism by treating it as something expressivism enables (or something like that; it's complicated.)
What's the difference?  It seems fairly easy to think of representations that do not express anything (I think), how is one to think of expression as not involving representation? 

A:

As Price acknowledges in Expressivism, Pragmatism and Representationalism, "expression" and "expressivism" are misnomers, simply used to label (multiple) alternatives to representationalism, "the assumption that the linguistic items in question 'stand for' or 'represent' something non-linguistic". This requires some sort of unmediated propositional access to said something, non-inferential cognitions through immediate sensations and/or intuitions. Descartes introduced the idea in modern times, and its criticism, along with the whole idea of "immediacy" in general, is usually traced back to Hegel. According to Miller, in Phenomenology of Spirit (§§90–110) Hegel argues that "any attempt to articulate, describe, or individuate the object of knowledge invariably makes use of either universals or contextually circumscribed indexicals (“this,” “here,” “now”) which necessarily require other items of knowledge, that is, they are epistemically mediated." 
In 19th century the argument was developed by Peirce, the founder of pragmatism, and through Lewis influenced Quine, Sellars and Davidson, possibly also Wittgenstein. On the continental side Hegel's argument was picked up by Heidegger, Merleau-Ponty and Adorno among others. Sellars elaborated it perhaps the most, and made its target famous under the name of the Myth of the Given. He and Wittgenstein especially emphasized how the myth conflicts with holistic and social aspects of concept formation, and both, but especially Sellars, are Brandom's inspirations. 
The pragmatist alternative to the Myth usually takes the form of inferential semantics, an elaboration on "meaning is use". At this point, even prominent reprsentationalists, like Fodor and Lepore, acknowledge (very grudgingly and derisively) in Reading Brandom that "it’s hard not to be impressed by the extent to which Inferential Role Semantics is the consensus view, not just in philosophy but also in cognitive science... It must be nice to have so many people on your side, but you don’t win a war just by assembling an army; you also have to win a battle or two."
According to the argument, impressions are not of a kind with representations, so identifying them is a category error, and there is an explanatory gap as to how one transmorphs into the other. Moreover, a representationalist faces an unpleasant dilemma: either impressions (like pain, etc.) have to be learned, or representations (e.g. concepts) have to be innate, for the two to "fuse" in perception. 
Here is Peirce from Four Incapacities (1868):

Every thought, however artificial and complex, is, so far as it is immediately present, a mere sensation without parts, and therefore, in itself, without similarity to any other, but incomparable with any other and absolutely sui generis. Whatever is wholly incomparable with anything else is wholly inexplicable, because explanation consists in bringing things under general laws or under natural classes... Finally, no present actual thought (which is a mere feeling) has any meaning, any intellectual value; for this lies not in what is actually thought, but in what this thought may be connected with in representation by subsequent thoughts; so that the meaning of a thought is altogether something virtual.

Here is Sellars from Empiricism and the Philosophy of Mind (1956):

"The notorious "ing-ed" ambiguity of "experience" must be kept in mind. For
  although seeing that x, over there, is red is an experiencing -- indeed, a
  paradigm case of experiencing -- it does not follow that the descriptive
  content of this experiencing is itself an experiencing... Certainly, the fact that something looks red to me can itself be experienced. But it is not itself an experiencing... It implies that while the process of acquiring the concept of green may -- indeed does -- involve a long history of acquiring piecemeal habits of response to various objects in various circumstances, there is an important sense in which one has no concept pertaining to the observable properties of physical objects in Space and Time unless one has them all - and, indeed, as we shall see, a great deal more besides."

