I loved DmC more than any entry in the previous series, and I was obsessed with the original when it came out. As far as the writing of the character goes, in my opinion Dante was always a douche, but in Ninja Theory’s game, he at least seemed like he could be an actual person rather than just a campy anime stereotype. The action in DmC felt modern, fun, and fluid in a way that the old games just aren’t. The levels and environments were way more dynamic and interesting than anything else in the series had been, and I loved that this game actually seemed to take place on Earth rather than some isolated spooky demon castle in the middle of nowhere.
I will say, though, most people that hate DmC seem to have a reverence for DMC3, which I just could not get into. It came out around the same time as God of War, and after playing God of War, DMC3’s action felt stale, dated, and punishing in ways that just weren’t fun. I even liked DMC4 quite a bit more than 3.
I’d love to see a follow up to DmC, but that seems incredibly unlikely, particularly how classic Dante is the character Capcom keeps putting in MVC…
ETA: To respond to the actual question in the headline, I think it can best be summed up by the original series was totally anime and the reboot was not.