I really liked the CV50 Strat I had years ago but the neck was skinnier than a regular Fender neck and I preferred a Fender neck. And ended up putting a Fender neck on it.
I'm tempted by the CV Teles but I have 3 Fender Teles and I don't know if the CV Teles are skinny, (I'm sure compared to my AO 50's most necks are skinny), so while I'm tempted by the CV Teles I always back out because I assume the neck width and nut width will feel skinny like my old CV Strat did.
I rarely get out to the stores and the last time I saw a CV Tele in a store it was a CV Custom Tele double-bound and it must've weighed about 12lbs. I couldn't believe how heavy it was. I don't remember how the neck felt because I couldn't get over how heavy it was.
But, seriously, they do have different neck profiles that you should consider. For example, I prefer the medium-jumbo frets on the Player vs the narrow-tall on the CV. There's also the 22 frets (Player) vs 21 frets (CV) and differences in the bridge design to consider.
I don't think they have a huge gap in terms of hardware or build quality, it's mainly differences in design and minor details. If you buy new from a store, it's more likely that the CV has had a lot more grubby paws on it than a Player and, as a result, may be further out of optimal setup.
If you are a new player, the CV is the best choice. If you are beyond the just learning stage, then the main difference is in the necks and pickups. If you find the CV neck to your liking, great, you have saved yourself $$$$
I’ve had both and the Squier was very heavy vs the Player and the neck on the Squier was thinner. I really love my Player but I bought it new when they were $699. If I was doing it now, I’d look for a lighter CV.