I am a member of both (and Sustrans)
BC for racing.
CTC for touring/leisure.
No problems with either, there is some cross-over these days. I support both as they both campaign and lobby for cyclists.
Both offer insurance cover, shop discounts, legal assistance (but this is just no win, no fee stuff so not that special...)
The CTC magazine is OK, a bit dated but it is trying to cover a wide readership.
If you are a racer/sportif/club rider, then go for BC
If you are a tourer/leisure rider, go for CTC