Aerial combos aren't exactly the most important of DMC3's innovations. But OK, I'll correct my statement; it seems to play exactly the same as a 11-year-old PS2 game, with exactly one innovation from the several that a 7-year-old PS2 game introduced.
Stuff that DmC incorporated from DMC3 which DMC1 didn't have:
*DmC has on the fly weapon swaps. Different inputs but similar application
*More moves for firearms outside of just shooting and charge shooting (Rainstorm, Fireworks etc)
*More mobility options outside of rolling and jumping (Air dashing, teleports)
*Enemy step can cancel air attacks (one of the major differences between DMC1 and DMC3)
*Stinger can be cancelled to Million Stab
*Style ranking system goes up to SSS and decays if you repeat moves. A progress bar is also displayed
*Similar checkpoint system from DMC3SE
*Gauntlet weapons now have a rising uppercut move (Magma Drive in DMC1 had no rising option)
*Multiple characters each with own playstyle/weapons
Probably a lot more stuff that I am missing. I think you need to go back to DMC1 and DMC3 to see how basic DMC1 was in terms of combat options. It prided in its enemy design and trained you to learn enemy patterns where as DMC3 trained you to learn real combos and push the combat system for maximum style efficiency. Style efficiency was irrelevant in DMC1 because as long as you don't get hit you will have a high style rank.
Some standard DmC combos