There is no real difference between them. The Classical approach as presented by dr. Tarrasch and a few other theoreticians is factly a misunderstood projection of the Steinitzian chess praxis. Nimzovich simply spotted and reverse-magnified the elements of the Tarrasch dogmas which were in contrast with the pure classicism.
Several contemporary strong players claim that they are classicists or hypermodernists, but they claim that tongue-in-cheek.
There seems to be a difference. The classical seeks to occupy the center for control. The hypermodern seeks to control it from afar and occupy it later. The games seem to take a drastically different turn depending on the style at play.
Which side are you on? I see benefits in both but I am leaning toward classical. It seems more direct and to the point.