With the risk of being redundant, I'll attempt to give my answer, if for no other reason than to help myself. Then again, the best people to answer this question are probably people who use but don't exclusively study mapping class groups.
The mapping class group is a group one associates to a surface, and it's true that it distinguishes two surfaces that are "visually different." However, I don't think this is where the true usefulness of it lies. For one thing, it is a group of homeomorphisms up to equivalence (isotopy), so studying the mapping class allows us to say that the homeomorphisms of two surfaces are different (or at least one of them has "more" or different relationships between them). This is still not the "best" use for the mapping class group though.
As mentioned in Matt E's answer, algebraic geometry is frequently interested in moduli space, and the mapping class group is the fundamental group of moduli space. Therefore it bridges between algebraic geometry and the study of surfaces.
But what it really boils down to is the mapping class group is a group of isotopy classes of homeomorphisms and will show up any time you want to discuss homeomorphisms of a surface and often when you want to discuss homeomorphisms of higher dimension and it's nice to have properties of such a group any time you want to talk about homeomorphisms. One thing I'm interested in, for example, is relating homeomorphisms of a base space and total space of a covering space and the mapping class group gives a language for this.
I think the Farb and Margalit book does a great job of motivating and in fact, is what first interested me in the topic.