What is the difference between "Sofa" and "Couch"?

They are effectively interchangeable in the US.

A Google ngram of the terms shows that in the US, couch was slightly more common until the middle of the 19th century, and the usage has been almost equal since then, with a very slight favoring of couch.

The ngram for British usage shows a similar pattern with equal usage kicking in about 1930 and a slight favoring of sofa currently.

Searches of the websites of two major US retailers of furniture (Macy's and Pottery Barn) for sofa and couch yield identical results in the number of hits, but the text describing the products tend to use sofa.

Though the two words are usually used interchangeably in everyday speech, there is a slight difference. Couches are typically armless, while sofas are not, and the distinction comes from their respective historical uses. Depending on the attributes of the furniture that you are referring to, it could either be a couch or a sofa.

If you'd like, you can refer to http://www.homedit.com/what%E2%80%99s-the-difference-between-sofa-and-couch/ for more information.

In Britain when I was young we had a couch but these days sofa seems to have taken over, and includes all sizes, with arms and without. Couch now sounds more formal to me, eg a psychiatrist's couch. Sofa-beds are also common, ie sofas which unfold/pull out to make beds for guests, whereas you don't hear of couch-beds. On the other hand you get couch potatoes not sofa potatoes.

I live in Western Canada, and here the words couch and sofa are in practical use, synonyms. I have older relatives who even still use the word chesterfield, although that word is decidedly more quaint and archaic.

This might be just me, but to my ear, couch is the slightly more informal of the two words. Sofa would be used if the piece of furniture is particularly more ostentatious, expensive or plush.