Reliance 606 natural gas water heater - model #640YORT - water does not stay hot. pilot has to be clicked to heat and then u can use hot water til tank empty and it will not continue to heat water.

If I understand correctly, the pilot light is not staying lit.
You light the pilot and water in the tank reaches temperature.
Once hot water is gone, then pilot light has to be lit again.

Gas pressure is good, and gas is flowing.
Thermostat is working at high end because it cuts off heating.
Vent is working because tank heats up.
Air intake is working.

Gas control thermostat might not be reading low temperature. Replace gas control valve.
Pilot tube might be dirty. Clean burner parts and combustion chamber, and air intake, and vent.
Backdraft might be blowing out pilot. Check for constant updraft by running water heater 5 minutes, and then light match under vent hood and see if smoke goes up.
Very hot air can put out pilot, especially when water heater located in attic. Add ventilation.
Improper vent might have water dripping back down into combustion chamber and putting out pilot light and rusting out the parts. Inspect for rusted parts and water.
http://waterheatertimer.org/How-to-install-gas-water-heater.html#troubleshoot
http://waterheatertimer.org/pdf/Pilot-light-will-not-stay-lit.pdf