I'm reading Milnor's Topology from the Differentiable Viewpoint and I've just finished the chapter about degree mod 2.
Since every two smooth homotopic maps $f,g$ must have the same degree mod 2 it's clear that we can assure that two maps can't be homotopic if the have different degree.
I'd like to see examples of some maps like that. Milnor's book is great but way too short!
An easy application of the $mod2$ degree would be that $\pi_n(S^n)\not \cong 0$:
Show that $id$ defines a non-trivial element by calculating its degree in comparison to the constant map.