classical vs modern

'The fact is that the Hypermodern Theory is merely the application, during the opening stages generally, of the same old principles through the medium of somewhat new tactics.
There has been no change in the fundamentals. The change has been only a change of form,
and not always for the best at that. ' - Capablanca

The simplistic answer is this.

Classical = Controlling the center directly with your pawns.

Hypermodern = Controlling the center from distance with your pieces - in most cases this means fianchettoed bishop(s).

Classical school was first, and hypermodern started emerging a little bit over 100 years ago.

Asking if one is superior to the other is probably similar to the question What is the best opening in chess? 

There are good positions in both types of openings and bad positions as well. That being said, hypermodern openings sometimes tend to be less straightforward than classical ones, so sometimes people don't recommend them to beginners. 

For a complete understanding of classical vs hypermodern approach , see the theoretical debate between tarrasch (torch bearer of classical school)and nimzovitch the founder of hypermodern approach alongside reti and Breyer, in fact hypermodern theory was born because there were already arguments in early 20 th century that chess has been played out there is nothing new to be played , it has become impossible for a master to win against another, new pieces must be added to make chess interesting again and stuff like that, so hypermoderns provided an answer by challenging the conventional wisdom and dogmatism of classical school by purposely playing openings which led to cramped positions this was a big sin according to classical players like tarrasch whose famed saying is "cramped positions have in them the germs of the coming defeat!"

You are welcome.