What are the advantages of the QGD over the Slav? I don't have an extensive knowledge of the opening theory of the two openings mentioned in the title. However, I do understand that, at a very basic level, in response to the Queen's Gambit the moves 2...c6 and 2...e6 have the effect of defending the d5 pawn as well as erecting a solid position for Black. 

However, the move 2...c6 has a simple advantage: Black's queen bishop is more easily developed. 

Why, then, is the move 2...e6 even played? What advantages does it have over 2...c6? It seems to me as though the move just impedes the development of a piece. 

The same consideration applies to semi-slav type structures in which the queen bishop is still on c8. Why would one want to play e6 before playing Bg4 or Bf5?
It is not so easy to develop c8 bishop to f5 or g4 in any of these systems. You run into many tactics like Nh4 winning bishop on f5 for a knight, f3 e4 as in open Slav or Qb3 Ne5 stuff with bishop on g4. The bishop is often developed to b7 in both semislav and QGD. It is safe and effective square for him. It is better to have the bishop passive for a while than to go for long term disadvantage of exchanging it for a knight.

That being said, e6 is good developing move even with bishop on c8. Now you can look for differences in 2...c6 and 2...e6 systems. The advantage of c6 is that it threatens to win a pawn forcing white either to close the c1 bishop (meran) or to be prepared to sacrifice it (botvinik, moscow). The advantage of e6 is obvious, the active c7-c5 will be played in one move, it doesn't take c6 square from knight and it helps you develop kingside faster.