ACS is the older organization, and NCS split off from them in the 1980s during one of the endless political fights that you get in organizations where the members are there to compete with each other. They're probably about the same in terms of respectability. I don't know whether one is bigger than the other, but FWIW the NCS Facebook page is more active than ACS's.
I get the impression that both organizations are struggling to survive. Aviculture in general is on the decline, and a lot of non-competitive bird clubs are going out of business. It's getting harder to put on bird shows, as animal rights organizations continue to chip away at our rights and more states are putting restrictions on public exhibitions and making it harder to bring birds in from out of state. Plus the clubs themselves continue to have their internal fights which doesn't make for a pleasant atmosphere.
I don't band at all. Banding is required to compete at the serious level at bird shows, but I don't do shows. If you're not into shows, the decision on banding comes down to weighing the advantages of being able to positively identify your bird if it escapes versus the risk of injury if the band gets caught on something. I think the risks outweigh the benefits so I don't do it. I do have an NCS-banded bird however (Frodo), who was given to me by a guy I know who was an ACS show judge.