I use Winchester brass for all but one cartridge I load for, the 6.5x55 gets Lapua. I find Remington too soft.

If your aren't gong to use Norma or Lapua, then its all about the same. If there is a difference between Rem and WW, I have not found it. Ku-dude

Generally Winchester is lighter and harder than Remington.
That being said does not mean that is always the case.
I do not use max loads so the difference in volume and hardness of the case head is of little matter to me. I suspect the Remington necks will take more loads before cracking but I anneal the necks of any brass if I get a few that crack. I use a lot of Win, Rem, and Federal brass.
While normally good, the worst lot of brass I ever bought was Norma. I have never used Lapua brass and do not know that I would gain much from its supposed better quality since I am not a competition shooter. I have heard good things about RWS brass but have never used it either. The Hornady brass I have reminds me of Winchester brass.
I usually buy once fired brass when I can get it in large lots with matching headstamps especially brass that was never on the ground. One thing I will say though is if you go the once fired route stay away from .303 British as it is often nearly separated on the first shot. I have also found that Remington factory ammo in 6MM Rem and 25/06 often produces blown primers and a lot of once fired brass in those 2 rounds will have loose primer pockets.