There are several advantages, most of which apply only marginally to bicycles. A nitrogen seller lists all of them, other sites list pros and cons. The key thing is that it's not about adding nitrogen so much as reducing oxygen, water and other gases.
Nitrogen molecules are larger than water, oxygen and most others, so they percolate through tyres more slowly. This means tyres stay at a usable pressure longer. This is the main benefit that a cyclist will see.
Nitrogen changes pressure with temperature slightly less than water does. But bicycle tyres stay close to ambient temperature almost always.
Nitrogen is less reactive than oxygen, so your tubes will last a little longer. But since tubes normally fail due to punctures rather than oxygen embrittlement it's going to be tricky to measure.
Note that the nitrogen in a bicycle tube will not be in contact with the rim, so the silly motorist argument about less rim corrosion doesn't apply to bicycles at all. In compensation bicycles get a much simpler change-over: just deflate the tube, squeeze all the air out of it, then inflate it with nitrogen. No need for repeated inflate-purge cycles.
But remember that the difference is only in the 20% of the molecules that you've changed from "other" to nitrogen, 80% are nitrogen in both cases.
Also, due to the difference in percolation rates, over time the air in your tyres will slowly become more nitrogen-rich (the other gases percolate out faster than the nitrogen). If you get no punctures, after a few years your tyre might contain over 80% nitrogen!