Apparently, not much! A peach has fuzz and a nectarine doesn’t. Nectarines seem to have a denser flesh than peaches, but they are very similar.
On a genetic level, http://www.michiganpeach.org/facts/nectar.htm states:
“The nectarine characteristic is controlled by a single recessive gene. In order for a tree to be a nectarine type, it has to receive the nectarine gene “g” from both parents. The cross between two nectarines will always result in nectarine offspring. Most peaches are homozygous for “G”, the peach characteristic. Two peach varieties that have a single nectarine gene are Autumnglo and Encore.”
On a cooking level, nectarines can be substituted for peaches and vice versa. Peaches, originally from Asia, are the “original” fruit and nectarines are a variant of the peach. Nectarines, though, are higher in Vitamin A and potassium than the peach. Both peaches and nectarines stand up well to cooking and canning. For jam, the use of pectin is recommended as they are not super high in natural pectins.
Peaches have two cultivars (or category types)–freestones and clingstones. Freestones are the kind of peaches that when you cut them, the separate from the pit easily.
Trees and fruiting are subject to frost which kills all the flower buds, and diseases such as brown rot, so they are a somewhat more high maintenance than other fruit trees and need full sun to grow/fruit.