I haven't played Civ VI for a while and so I know there's some major patches I haven't played with yet, so my opinion is based on an earlier build of Civ VI. But I like both, and going forward with future expansions I suspect Civ VI will be my favourite. Right now Civ V is definitely a bit more robust, which makes sense since it's a game fleshed out with two major expansions.
Civ VI in comparison to Civ V I feel has weaker AI particularly in military engagements.
I love the district system, although I don't like the way production costs scale further into the game so new districts in late game cities become incredibly slow to get up and running. I also don't like how that forces a bit of extra min max micromanagement for me where it becomes important to place districts as soon as you have the population for it, even if you don't intend on building it until later. I don't like how aircraft require their own district though and for me it makes it not worth investing in over other projects. I think some district buildings are a little overpowered too, like the production buildings that give +production to all cities within 6 tiles of each other.
I love the civics/government system with cards as it varies the gameplay and encourages me to commit to a plan ahead of time, maximising the use of card bonuses. I also like the boosts to tech and civics through gameplay actions, although I dislike the RNG inherent in them as sometimes you can't take advantage of them due to things outside your control.
I dislike the lack of World Congress or something similar to it. I also dislike the religion part of the game and it feels completely irrelevant to my games. However I realise this has received a major overhaul since I've played.