What is the difference between Nm3 and Sm3?

Unfortunately neither Nm3 (normal cubic meter) or Sm3 (standard cubic meter) are complete definitions in themselves. It is essential to know the standard reference conditions of temperature and pressure to define the gas volume since there are various debates about what normal and standard should be.

Most commonly used reference conditions are:

Normal cubic meter (Nm3) - Temperature: 0 °C, Pressure: 1.01325 barA
Standard cubic meter (Sm3) - Temperature: 20 °C, Pressure: 1.01325 barA

barA: absolute pressure

How do I calculate Nm3 and Sm3 and what is the conversion rate?

The volume of gases changes with temperature and pressure, therefore these parameters are also part of the conversion equation.

The conversion from Sm3 to Nm3:

V1/V2 = (P2xT1) / (P1xT2)
V1/V2 = (293.16x1.013) / (273.16x1.013) = 1.0732

Temperature is entered in K; 273.16 is temperature of tripple point of water.

Interpretation: Gas of certian weight takes under normal conditions volume of 7,32% smaller than under standard conditions. Therefore 1 Nm3 contains 7.32% more gas than 1Sm3.

Conversion for reference conditions 15°C a 981mBar:

V1/V2 = (288.16x1.013) / (273.16x0.981) = 1.08932389

For the completeness' sake we state the comparison on the weight basis:

Oxygen:
1m3 (Temperature: 0 °C, Pressure: 1.01325 barA)  weights 1.43kg
1m3 (Temperature: 20 °C, Pressure: 1.01325 barA)  weights 1.33kg

Nitrogen:
1m3 (Temperature: 0 °C, Pressure: 1.01325 barA)  weights 1.25kg
1m3 (Temperature: 20 °C, Pressure: 1.01325 barA)  weights 1.16kg

Assumption: It is essential to consider these facts when projecting the gas generating system or when actually making decision to purchase certain model because you might be actually buying less than you actually think you are.