NAUI requires you to swim, although there is no specific distance requirement. PADI allows you to snorkel instead of swimming. PADI requires 200 yds if you swim, 300 yds if you snorkel.
NAUI requires either 4 SCUBA dives and 1 skin dive or 5 SCUBA dives. PADI requires 4 SCUBA dives.
A NAUI instructor can have no more than 10 students in open water regardless of how many AIs or DMs he may have assisting. A PADI can have 12 with 2 assistants.
NAUI requires an underwater swim of 50 ft. PADI does not.
NAUI requires you to breathe through a flooded snorkel. PADI does not.
NAUI requires you to orally inflate your buddy's BC. PADI does not.
PADI requires no mask breathing. NAUI does not.
NAUI requires you to remove a simulated cramp from your buddy. PADI does not.
PADI requires breathing from a free flowing regulator. NAUI does not.
PADI requires you to disconnect your LPI on the surface. NAUI does not.
PADI requires snorkel/regulator exchanges. NAUI does not.
NAUI requires simulated rescue breathing. PADI does not.
NAUI requires bringing a simulated unconscious victim from 10 ft. PADI does not.
NAUI requires students to figure air consumption. PADI does not.
PADI requires students to learn buoyancy control using oral inflation. NAUI does not.
NAUI allows its instructors to add requirements. For example, while NAUI does not require no mask breahing, I've never heard of a NAUI instructor who didn't require the skill. PADI does not. While a PADI instructor can teach additional material, he cannot require you to learn it. For example, a PADI instructor could decide it's a very good idea for Open Water students to learn to rescue an unconscious diver from the bottom of the pool and he is free to teach the skill. If a student doesn't want to learn that skill, the instructor cannot refuse to certify the syudent on that basis.