Seeking an intuitive explanation of the Mapping Class Group For a surface $S$ the mapping class group $MCG(S)$ of $S$ is defined as the group of isotopy classes of orientation preserving diffeomorphisms of $S$:
$$MCG(S)=Diff^+(S)/Diff_0(S).$$
I understand this definition as well as all of its component pieces. What I don't understand is why this quotient is a natural thing to study.
Specifically, I can see why the full diffeomorphism group $Diff(S)$ would be natural to study, and if $S$ happens to be orientable, I can see why it would be reasonable to restrict ones attention to $Diff^+(S)$. However, I don't see why the quotient is a natural or intuitive next step. Is there a good explanation why diffeomorphisms that are isotopic to the identity are 'uninteresting'? Thanks!

If you view an isotopy as a path in the space of diffeomorphisms, each element of the mapping class group corresponds to a path component of the orientation-preserving diffeomorphism group.
