I have done both.
In England, I don't use their biking maps (they have several groups that produce them). Instead, I get a country motoring atlas (1 inch = 3 miles) and simply plan my route trying to use the most local roads I can. I probably miss some things but I really like finding my way on my own.
In the US, I use the Adventure cycling maps. I like the information they provide and the routes they choose. That said, I always check them with other resources and change them if I see a better way.
If you are following an Adventure cycling route, why not use them as a resource?
Ray