I like both and have owned both, but other than the stainless metal are about as opposite as you will get in 2 rifles. Both will be quite accurate with a slight edge going to the Tikka in my experience.
The Weatherby is probably the heaviest rifle made and is very conventional in looks and operation. If you think you may want to in the future, there are many more options for aftermarket stocks etc for the Vanguard.
The Tikka is the lightest rifle you can buy for under $1,000. They have great triggers and are usually one of the most accurate rifles out there. Many don't care for the unconventional looks and find they are just TOO light for their tastes. While the guns themselves are very accurate, many guys cannot shoot a rifle that light accurately from field positions. Some guys like the detachable mag, others don't. Either way if you want another mag they are quite expensive ($60-$70) and can be hard to find.
Both are good solid rifles in my experience. You just have to decide which features are most important to you.