There isn't necessarily "extra" sidewall in either a 15 or 16 inch tire. It depends on aspect ratio and tire size just how much sidewall you have. One advantage to 16's, all other considerations being equal, is there is a somewhat greater choice of tall skinny tires in 16. One such example would be 235/85/16. I don't find anyone who makes 235/85/15. An extreme example going the other way would be to look at 20 inch tires with an aspect ratio of 45. These tires have nearly no sidewall despite being much taller tires. If, what was meant, was that for the same diameter tire (such as a 32 inch tire), a x15 inch tire will have more sidewall than a x16, this would be true because the height has to come from somewhere.
Generally, you can change from 15 to 16, but not the other way around if your vehicle was built to take a 16 because the smaller wheel won't clear the brakes.
The original question was with regard to the advantage of running 16 versus 15. I believe the advantage is in the availability of choice in tall skinny tires, which are the most popular style for expedition use.