Six of one, half dozen of the other. I'd buy the one that offered the features I wanted in a chambering I desired. I don't think you will go wrong choosing either rifle.
10 years ago Remington was building a better product. Today Winchester is building a better product. 10 years from now who knows.
I've always owned and prefered Remingtons, but I can't think of a single rifle they make today that I'd buy. Not that they are poorly built, I just don't like the current offerings. The new Winchester Featherweight and Extreme Weather are exactly what I want a rifle to be.
Between the two, I would choose Winchester. Reason, my economy priced old Winchester Ranger in 30-06 performs flawlessly, shoots sub .75" five shot groups at 100 yards and didn't cost much. What more could you ask?
Between the two, the Model 70s that I've handled that FN is now making appear to be very nicely constructed. Recent Model 700s, not so much. My personal choice is the sturdy Ruger Model 77 (either the originals or the later MKIIs). I prefer the Ruger's looks, the Mauser type extractor and its handling characteristics. All the ones I have are plenty accurate.
I faced this exact choice two years ago and bought a new Winchester M70 Featherweight. I liked the smoothness of the action, the handling, and the wood finish and metalwork just looked nicer on the Winchester. I have to say that I'm VERY pleased with the M70.
If I were buying a "new" hunting rifle today, it would be a used one. Tons and tons of fine, used rifles out there, in all calibers and configurations. A minor cosmetic ding, a handful of rounds fired (seldome more that that), and lots of $ saved for a rifle that will perform in the game fields exactly the same a one new out of the box.
Older Remingtons and Winchesters are generally good guns, some models a little better than others. New ones? I don't have a clue. Sorry.
I'd go with the Model 70, but I've always considered myself a Winchester guy. Had a Remington 700 Mountain rifle in .280 Rem. when I was younger which was stolen when our house was broken into, they also got a post 64 Model 70 Varmint in .225 Win. Of the two I miss the M70 more, not that the Remington was a bad gun, just preferred the M70. The one thing that always irritated me with the Mountain rifle was sighting it in seemed to take forever, 3 shots out of that rifle and they'd start walking off the paper due to the barrel heating up. I would own another Mountain rifle in .280 Rem if I got one for the right price, but I'd prefer a M70 too bad they don't chamber for .280.
If I was buying a new one right now it would probably be a Model 70 stainless featherweight in maybe a .308. I handled one of those the other day and it was sweet.........