Everyone that I know with a Remington 700 in some flavor of tactical or varmint model can get sub MOA out of their rifles. I recently got into bolt guns and since having gotten into them, I've been taking note. What I've seen is that they all shoot good.
The Savages are arguably more accurate, due to the design of the bolt, but Remington bolts can be doctored up. Remingtons are arguably stronger, due to the design of the bolt. I don't know first hand, but I've also been told that you can't bed a Savage stock due to the way it is inletted. I don't know if it's true, but perhaps something to check on. On the other hand, the same guy told me that you really don't need to bed a Savage stock because you can get super good results by just fiddling with the amount of torque that you put on the action screws. Find what it likes and use a torque wrench to repeat.
Since I got a Remington that shoots really well, I'll say Remington. In the end, you'd probably be happy with either a Savage or Remington. I do know a guy that regularly shoots on the same days that I do at the range and he has gotten to be a Savage freak. Every other week he has a new Savage telling me about the deal that he got on gunbroker on the used rifle. He swears that they shoot as well as any custom job he's ever had. Of course, they are all heavy barrel models.
If you're on a tight budget, check out the Stevens rifles. I hear that they are great shooters.