Why use liquid nitrogen to make ice cream?
Ice cream is an emulsion and in it, you have air, fat, and ice.  The smaller the ice crystals, the smoother the icecream and better chance of achieving a velvety, creamy, smooth texture. The flavor is also elevated given the smaller crystals. Put it in the freezer, and ice crystals start to grow bigger and you lose the benefits. 

Also, colder temperature inhibits the taste buds ability to pick up sweetness.  At liquid nitro cooled  temperatures, you can get an explosion of flavour as the icecream warms up and melts in your mouth and the taste buds start to pick up on sweetness. 

Finally, aside from a show, it's much much faster to make icecream with LN2 than in a regular machine. Seconds versus 30-45minutes. 

There are hazardous issues with liquid nitrogen and handling. As a starting point, look into this primer.