Both are good rifles. I have a Rem 700 30-06 that I bought new 38 years ago, and it still shoots good. I have also shot my Dad's .300 Wby Mag, which is an old model Vanguard. It shoots one inch three shot groups. Here is my opinion, realizing they are totally subjective:
Advantages or better features on the Vanguard.
1. Shrouded bolt with side gas relief holes.
2. Monte Carlo style stock. For me, it feels and shoots more comfortably.
3. Safety locks the bolt. I happen to like it that way.
Advantages or better features on the Remington, keeping in mind it is my old one which may differ from some of the new ones.
1. Crisp trigger that breaks like a glass rod.
2. Faster lock time
3. Shape of bolt handle (again, for me)
Disadvantages of Vanguard:
1. A terrible trigger. It was gritty, heavy, and inconsistent in let off.
Had a gunsmith work it over, and now it is very good but not as good as a 700 trigger.
2. Safety is a sliding type on the right side. Personally, I do not like it because it is not positive enough. I always think it would be easily bumped forward to the fire position.
Disadvantages of Rem 700:
1. Safeties on new ones, at least the last ones I saw, do not lock the bolt. On a hunting gun, this can cause problems of brush knocking the bolt open and dumping a round onto the ground when slung on your shoulder.
2. Most of the new Rems I have seen all have classic style stocks, which simply seem to pound me more than the Monte Carlos.
Hope this helps. I understand that the new Vanguards have hand honed triggers and are dramatically better in that respect.