Hey, that's fun, I'll do it, too!
Better Main Quest: Can't comment on either yet, as I'm far from done with them
Better Main Cast: FFXV
Better Female Cast: TW3
Better Sidequests: FFXV, actually. Now let me explain myself on this, since this is most likely controversial. What I care about the most when it comes to sidequests is the core gameplay loop and the actual quest design. When it comes to the quest design itself, I don't think either is good for the most part. Witcher III has amazing context developed through dialogue and just general writing for its quests, but the mechanical part is no different from FFXV's fetch quests. In fact, I'd say it's worse due to over reliance on witcher sense. At least to catch the dumb frogs in FFXV I have to listen to their sound and follow it, instead of pressing my detective button and follow the red line.
The core gameplay loop is just more enjoyable to me in FFXV, since I prefer its combat as well as its exploration, and I'd consider the optional dungeons, hunts and etc side content as well, and one that can't be ignored. There's nothing like finding an optional dungeon with nice, connected level design and a good challenge in Witcher 3. The caves there just exist for lore purposes, those beasts need somewhere to hide, but there's no fun to be had in exploring them.
A game that did quest design better than both was Skyrim, to be honest. The way you start the Dark Brotherhood questline is amazing, and more than just someone telling you a sad story while the wind blows indoors, or someone telling you to pick up some random shit because reasons. To me, what you're actually doing in the quest is the quest design, not just the context established through writing.
Better Writing: TW3
Better Mini Game: Can't decide on Gwent or Fishing. Probably Gwent.
Better World: FFXV. The world in The Witcher is better realized, for sure, but I tend to prefer the more fantastical terrains and enemies of Eos. Also really enjoy the backdrop of the Starscourge. I also really enjoy the way FFXV handles open world. With meaningful content like well designed dungeons here and there instead of a bunch of stuff to do in the overworld itself but with little to actually find through exploration.
Better Music: FFXV, but as you said, they're both REALLY good. I tend to prefer the occasional overworld music playing over it being constant, though, so when Wanderlust plays, I'm super excited to hear it again, instead of tired of listening to it for hours and hours.
Better Artistic Direction: FFXV (animeeee)
Better Character Movement: FFXV, but this is a weird category. Do things like warping and shit count or are we just talking about basic movement? Because if they do, then it's not really fair to compare.
Better Menu Design: Neither is super remarkable, imo. I enjoy the UI of both, but the menus are "ok". The menu music in FFXV is sick, though, so I'll go with that. FFXV.
Better Fashion: Prompto. Would be Noctis if his casual clothes weren't shit. Would be Geralt if we had more casual clothes with useless stats just to use in Novigrad when walking around. I'm the kind of idiot who unequips their armor when swimming.
Better Hair: The White Wolf. Ok, I'll have to stay with you, just because of the customization options. The amount of times I went back to the city just to get a haircut or beard change... hahaha. Noctis' hair is fantastic, though. I prefer it to any of the hairstyles Geralt can have, but alone it can't match changing it for whatever stupid reason you have.
Better Otaku Pandering: FFXV. Y A O I B O I Z
Better Awkward Sex Pandering: TW3. I mean, it's the only one with sex. I think?
