Let's be getting into details.
PROS:
Limbo - I really like the aesthetic and feel of Limbo. The garish colours standing in sharp contrast to the dull grey of the real world, the random patches of black appearing, the way ordinary objects twist and turn when you go near them, and the way the enviroments suddenly change all give it the feeling of this being another world, but at the same time not. I also kind of like the idea that Dante can only access his powers while in Limbo, due to it not being the normal world.
Combat - The combat is great, in that it flows well and actually rewards altering your tactics a bit. At first it took me some time to adjust to the fact that there was no Lock-on button, but once that was done I was free to slaughter anything in my way, evoking the same feeling the previous DMC-games gave me.
Vergil - I actually really like this new version of Vergil. As I see it, he's very much the opposite of Dante in personality, but having roughly the same goal. Dante is very brash and outspoken, but would never leave anyone to die, while Vergil is very calm and collected, but doesn't even bat an eye at the thought of sacrificing someone else for the greater good. Unfortunately, this is all rendered moot by the last fudging level...
CONS:
Combos - There seems to be less different combos present in DMC: The Reboot. And depending on your viewpoint, this can be either good or bad. On the one hand, it makes the game easier to play, especially for people new to the series. On the other hand, it means you spend the whole game churning out the same seven combos over and over, because there are no more combos after that. Not a great con, but worth bringing up nontheless.
Dante - Okay, I for one don't particulary mind the new design of Dante, even though I prefer the old one. What I'm talking about here is his personality. I'm not about to complain because the new Dante is a smug douche, because let's face it, Dante has ALWAYS been a smug douche. But in the previous games he looked like he was having fun, and him being a douche was just a way to reinforce that. The new Dante feels more like he is a douche simply for the sake of being a douche. Also, what is up with the new design of his Devil Trigger? His hair turns white? Wow. Spectacular.
Nephilim - Okay, I like the idea of opposites making out and spawning offspring as much as anyone else, but the fact that Dante and Vergil where part human part demon used to have some kind of point beyond "they are awesome and the best of both worlds". Most demons they encountered were basically mocking them for being half-bloods, not realizing that the brothers were probably the most dangerous individuals out there. Also, it fit into the roles of each brother. Dante chose the path of the humans, and used his powers to protect them from the demons, whereas Vergil embraced his demonic heritage and attempted to gain more and more power through that. All of the above is, so far, completely thrown out the window in the new game, and them being nephilim has no impact on their characters beyond "they are the chosen ones" and "Vergil is a douche". Speaking of...
The Final Battle - Okay, so I should have suspected that something was up when the battle against Mundus took place on level 19, when these games have always had 20 levels. And yeah, the battle against Vergil is pretty awesome, but... it's just so stupid. It comes completely out of the left field, with Vergil suddenly proclaiming "MUAHAHA! I ARE GOING T00 TAEK TIS W0RLD OVA, LOL!" at the beginning of the level, and then you fight. And this is the first time in the entire game that it is even hinted that he wants to take over after Mundus (okay, some people in the game points out that once Mundus is gone, who will take over, but I just thought that was their way of saying "there will always be demons"), after you have spent a good part of the game working together with him to take down Mundus. Also, he suddenly has a great disdain for humans that wasn't present just a few minutes ago. It feels like they decided to put this fight in just for the heck of it. I mean, in DMC3 the final battle against Vergil makes sense. You have already fought this guy twice, and the only reason he hasn't scewered you already is that you were both preoccupied by this other douche trying to kill you. Also, in DMC3 Vergil was made out as a villian from the very start, not just in the last few minutes.
Mundus Was Right - All of this Vergil malarckey would probably be fine, if it hadn't been for one small tiny little detail: I actually agree with Vergil, and by extension Mundus. Humans are not fit to govern themselves on a global scale. Sure, Mundus's worldwide 1989-state was a bit extreme, but he points out that humanity would probably have been in worse shape had he not taken over and straightened things out. Dante spews out some bullshit that the people should govern themselves or some crap like that, but he doesn't seem to grasp the consequences. This DMC-world, for better or worse, has spent the last decades under the control of one single individual. And while that individual oppressed everything and everyone (speaking of which, the anti-consumerism in this game would be less blunt if a parade rolled by trumpeting it out), he also kept things running smoothly (after all, there is no mention of any war or conflict besides Mundus vs. Dante/The Order in the entire game). What do you think will happen when that one individual is gone? I don't think humanity is selfless enough to just work it out together. I expect major chaos and riots all over. On top of that, you assholes managed to break the border between Limbo and the real world, so now demons are roaming around in broad daylight, no doubt slaughtering countless individuals while doing so. Nice job guys. Hope you enjoy your freedom, in the few moments before demons eat you all of for breakfast.