Why do we care whether a functor is representable? In the algebraic geometry textbook by Görtz and Wedhorn, the authors prove that several common functors are representable. For example, the Grassmannian functors are representable. 
Beyond being cute category-theoretical facts, why does this matter? What practical benefits come form knowing that certain functors are representable? Does this observation simplify or make possible the proofs of certain theorems, for example? This is perhaps too broad a question, so I should mention I'm mostly interested in the context of algebraic geometry.
 A: Functors are hard. Schemes are much easier. If a functor $F$ is representable by $X$ then it's values at $T$ is completely described by maps from $T$ to $X$. For example, for a field $K$, you get a bijection $F(Spec K) = Hom(Spec K,X)$, so $F(Spec K)$ is in bijection with the $K$-points of $X$, so you can study your abstract functor by studying the scheme, which is much easier to study - the theory of schemes is very well developed. Far less can be said about general functors. For example, schemes can be described by equations. If you can find equations for your scheme then that will often tell you a lot of information about the stuff your functor is parametrizing which otherwise would be very difficult to obtain. Some other useful aspects: the dimension of the representing scheme tells you roughly how many ``parameters'' is needed to specify an element of your functor. If $X$ is a curve then the genus roughly tells you how many $K$-rational points it has if $K$ is a global field.
A: There are (at least) two aspects. First, a functor which is known to be representable, has nice categorical properties, for example representable functors are trivially continous, that is they preserve all limits.
But if we know an isomorphism $F\cong\hom(X,-)$, this does not only tell us something about $F$. It tells us something about the representing object $X$. In fact, by the Yoneda lemma, it tells us everyting about the representing object $X$: An isomorphism $\hom(X,-)\cong\hom(Y,-)$ yields a unique isomorphism $X\cong Y$. This kind of reasoning is used all the time in algebraic geometry (and everywhere else in maths).
