My Cavern and Cave I certs were through NAUI. NAUI cave I is roughly equivalent to Apprentice (dive to 1/3rds, two jumps) except NAUI Cave I is no deco and Apprentice allows limited deco. I think NACD and NSS-CDS intro are both 4 dives and apprentice is 4 dives; NAUI Cave I is 10 dives. Apprentice also expires in a year; NAUI Cave I does not. The friend I took NAUI Cave I shortly after has gone on to take full Cave, through NSS-CDS I believe, and her instructor accepted NAUI Cave I as Apprentice, although, as others have mentioned here, I'm sure it was skill level, not ownership of a cert card that was the deciding factor. I'm diving for a year or 15 months at NAUI Cave I before I take my next training, which I expect to be Cave, probably from a NSS-CDS instructor. There's a lot of cave to see within the 1/3rds of doubles and two jumps limitation -- in fact, my first couple of months of diving after certification I didn't leave the gold line -- just because my card said I could do it didn't mean I had to. I've also had the good fortune to dive with a number of very experienced cave divers who have been very generous with their time and knowledge; I will be much more prepared for my next class because of this informal mentoring/training/experience. I do plan to use a different instructor for my next class, not because there's anything wrong with my original instructor, but because each instructor, while teaching the same basic skills, has their own way of doing things, and seeing things done multiple ways is helpful. I've been a snow skier for many years and take lessons every year and every year I use a different instructor because, while they all teach the same skills, they all do it in a different way, and multiple perspectives help me learn. So don't worry about the issuing agency, get a good instructor.
I am not advocating NAUI here; it's just worked well for me and I think makes a good comparison case for PADI Cave training. No one has yet refused to dive with me because my card has NAUI on it.