NSRA didn't used to include cover for rough shooting but it does now. If ever you want to take part in NSRA sponsored paper punch disciplines at national level you'll need NSRA membership and you'll also need it if you want to become a club instructor. So there are good reasons to see NSRA as the first option.
If you get BASC and then need NSRA you are paying twice for some aspects. You could perceive that BASC does more in the way of advice and PR for shooting; BASC is quite oriented to shotgunning and NSRA to small bore, as the name implies. Neither of them in my opinion majors on air gun shooting. It has helped though, since BFTA handed its training and qualification processes to NSRA for administration.