First, please understand that there are more players than just NAUI and PADI. In fact, of all the major players, PADI is the most recent.
For the record, John Cronin, the former NAUI instructor who started PADI, has recently died. He was a great pioneer in recreational scuba diving, and he built a world-wide empire unrivaled by his predicessors in NAUI, YMCA, or SSI. Enough said.
Second, please note that there are many instructors currently in NAUI, YMCA, SSI, and PADI that cross-certified from other agencies. And that really muddies the difference between them.
Ultimately, your diving skill will depend on your specific instructor, not the agency he/she is certified with.
Ultimately as well, the quality of your gear and your diving opportunities will depend on the dive store that you will frequent.
If I myself were rating these particular diver instruction programs, I would rate them like this:
NAUI
SSI
YMCA
PADI
But they each have their pros and cons.
First, gear. Where I live, the very best gear is available at an SSI store. That store also has their own on-site pool. That store and their instructors are just as good as any other store around, but their gear is probably superior. They offer ScubaPro, Zeagle, Sherwood, Oceanic, and more. That is practically the whole spectrum of gear. They do a lot of business, and so they can afford to carry a lot of different lines.
The main danger to avoid is from a store that only offers a few lines, and then they pressure you into buying their specific brands of gear alone, without giving you a choice. And if they dont have an on-site maintenance technician, then they can be a hazard to you, if they sell you gear that you cant easily get repaired or maintained locally.
Second, instructors. I classify SSI and YMCA instructors as fairly similar. They both use a fairly rigorous fixed program for training, and they both work hard and give you a lot of diving time to train you. Thats not to say that PADI instructors dont. There are a lot of extremely good PADI instructors out there teaching. Some of them even crossed over to PADI from NAUI or SSI or YMCA.
So what about PADI? Their main advantage is that they are the giant, especially internationally. When you go to book your international travel, you will have more resort choices if you book through PADI. Ouch! I hate that! But its true. And dive travel is a big part of scuba diving.
PADI's best instruction is in their Rescue class and also in their DiveMaster program. Nobody teaches a better rescue class than PADI. And nobody's divemasters are better trained than PADI's.
Finally, what about NAUI? We tell people that WE are the best training agency. We take great pride in that. Then we attempt to prove it by customizing our dive traing for each individual student in our classes. And we give them an many dives as they need to get them where they want to be. I am biased, so dont take my own word for it.
You have to choose for yourself. Shop around. Take a few courses from each. Then make up your own mind for your own reasons. It will mostly depend on the local instructors and the local stores available to you. Only you can find out who is the best in your neighborhood.