graphs where vertex coloring is in P but independent set is NP complete
Supposedly the reference "NP-complete problems on a 3-connected cubic planar graph and their applications" by Uehara (a paper I haven't actually seen) proves that independent set is NP-complete even for triangle-free planar graphs. But by Grötzsch's theorem they are always 3-colorable, and testing for smaller numbers of colors than 3 is easy in any graph, so they can be optimally colored in P.

Circle graphs have the opposite property: for them, coloring is NP complete but the independent set problem is easy.