A couple of things: The NAUI tables are the Navy Tables with one step backed off to be a bit more conservative. That's why the Navy tables allow 60 feet for 60 minutes but the NAUI tables go for 55 minutes. There are now some modifications on ascents thrown in for a bit more good measure.
The PADI tables were based off of the same theory a la Haldane (tissue M-value based). However, the consideration of PADI was that these tables are built for recreational diving, which does not allow for mandatory decompression and calls for remaining within certain time and depth limits.
Getting into theory (ho-hum for some of you out there), the Navy and NAUI tables use a slower compartment for controlling the maximum dive times and tissue saturation levels in determining depth and time limits. These were, at the time of the construction of the tables, thought to be the slow tissue compartments which ongas more slowly and offgas more slowly, too.
The conclusion of PADI was that recreational divers do not do the kind of diving that makes this slower compartment get saturated due to the shorter dive times and shallower depths carried out by recreational divers. So why base diving off of that compartment?
PADI uses a faster compartment (60 versus 120 minutes) for the controller in its table formulation. As a result, this tends to make most initial dives shorter. However, inasmuch as the diver will theoretically offgas faster between dives in this compartment, it allows for shorter surface intervals and somewhat longer (depending on profiles) second dives.
Of course, time and theories have advanced since the creation of both of these tables. More and more divers are using bubble theory incorporated in some measure or other in their computation algorithms. That's why you will find that most technical divers opt for software that incorporates this in its calculations (such as V-Planner) rather than using dive tables (other than the new NAUI RGBM based tables). It is also why you will find that these divers tend to start their safety/decompression stops deeper than with conventional tables and why their "no stop" times tend to be even more conservative than with these tables.
If you have not been diving using this type of planning yet, you will notice a difference in how you feel when you exit the water after a few hours. That pretty much says it all in my opinion.
FWIW, I was a Navy table diver since the 80s and made the switch to V-Planner about a year and a half ago. I still carry my Navy Tables with me in case I have to use them on the fly. I have learned the patterns of my software and can make a quick and dirty conversion from the Navy tables in an emergency if I have to (but I always use my software for planning).
Back to the subject at hand, I took training from several different agencies including NASDS, PADI and NAUI before I became an instructor. I absolutely noticed a difference between what the agencies teach and how they teach it.
For that reason, even though I was already a PADI Divemaster, I elected to become and NAUI instructor and later went on to become an Instructor Trainer and then a Course Director. I actually had to go back two steps to become a NAUI instructor when I just could have taken one step to become a PADI instructor. But I was doing this for reasons other than just getting certified. There are differences between the agencies.
That being said, I continued to train with other instructors of other agencies, including PADI, because each agency does have some exceptional instructors. The instructor is the most important thing but there are differences between what each agency does and how it does it. I feel that the only way for anyone to understand the differences is to take training from each and then make the decision.
