(I apologize that this is a vague question).

I seems to me somehow that homotopy groups behave well with respect to (Serre)-fibrations. For example you get a long exact sequence of homotopy groups from it. On the other hand cofibrations and homotopy groups seems to be no good friends at all (e.g. $S^1\to D^2\to S^2$).

But then again, the situation in homology seems to be the other way round. They behave well with respect to cofibrations (you get a long exact sequence) and fibrations are harder to investigate (Serre spectral sequence etc.).

My question is: What is the intuition behind this difference? (in particular with respect to the fact that homology is just homotopy of another space.)