As an inert gas, nitrogen prevents oxidation of the cut surface, helps clear the cut, and reduces or eliminates slag. For most people, that makes sense - as in welding, we use an inert gas to keep oxygen off the cut. In contrast, the use of oxygen might make less sense. Why use one gas that prevents oxidation, and then another that basically guarantees it?
Unlike nitrogen, oxygen assist is used as an oxidizer to assist in the vaporization of the material. The oxygen doesn't burn (on it's own, oxygen isn't flammable!) But pure oxygen mixed with, say, metal dust, absolutely does burn. Injecting oxygen into the cut effectively adds energy to the process, allowing thicker materials to be processed. Trumpf - the manufacturer of our laser - indicated that oxygen gas adds on the order of 3 kW to the cut power, doubling the effective output on our machine.
Thick steel, aluminum, and even relatively thin copper require oxygen assist. Thinner material can be cut with either oxygen or nitrogen, depending on the desired surface finish. As shown above, oxygen cuts leaves a glassy scale layer on steel. It actually looks good, but it won't hold paint. Powder coating processes won't work on it, either. In contrast, a nitrogen-assist cut comes off the machine ready to go.
For that reason, nitrogen is usually the preferable choice wherever it is an option. The tradeoff is part cost. On our machine, 1/8" to 1/4" thick steel requires a lot of nitrogen - about 42 sfcm, which will empty a 230 liter liquid nitrogen dewar in an hour and a half. Cutting 1/8" steel with nitrogen costs around $1.13 per minute, for the gas alone! In comparison, cutting the same material with oxygen costs around $0.13 per minute for gas - less than one tenth the cost.
The cost tradeoffs aren't always straightforward. Post operations cost money, too, and on our machine, nitrogen allows for cutting speeds 71% higher (in 1/8" steel) compared to oxygen. The best choice of gas will depend in large measure on the specific part, and what post operations will be required.
Because of the high gas cost, however, we use oxygen for steel thicknesses of 1/8" and above, unless the customer requests nitrogen specifically.