Comparing sub-zero temperatures to liquid nitrogen temperatures doesn't quite do justice to liquid nitrogen. Imagine how hot boiling water is, and how cold ice is. Think of the temperature difference between the two. Now, take a step of that same size, colder than ice. Now, take a second step. That's how cold liquid nitrogen is. Liquid nitrogen will be at -196 degrees Celsius. Even Antartica, at the coldest temperature ever recorded, only got down to -89, still nowhere close to LN2.

The problem with the liquid nitrogen enhanced drinks is when people drink them before the LN2 has evaporated. You then have that -196 degree liquid in your stomach, and that's what freezes sections of it. Safety regulations require that the bartender not give you your drink until all the LN2 has evaporated.

I used LN2 to cool gamma-ray detectors for a while, and it is really fun stuff. Like driving a car or cooking raw meat, you just need to know what the dangers are and how to avoid them.

I think there is quite a difference between a shot of sub-zero alcohol and a ingesting liquid nitrogen, which is at minimum -210°C vs the at most -70°C the alcohol will have, though likely the easier drinks will freeze at higher temperature anyway, usually starting at -5°C for a beer.

edit: It also helps that ethanol is likely going to stay liquid in your stomach, liquid nitrogen boils off; that pulls a lot more energy from the environment.