Peaches and nectarines are so closely related that if you plant 100 peach pits, a few will grow into nectarine trees, and vice versa. It's a fun fact, but mainly for botanical geeks.
There is a practical application, though, and that is that you can use the two kings of summer fruit pretty much interchangeably in recipes. The results will be slightly different, but not in a bad way. To me, nectarines (the hairless ones) have a little bit lighter, slightly brighter, more lemony flavor. Peaches (hairy, or "pubescent" as the plant breeders say) seem richer and muskier.
Used in recipes, the difference is mainly one of tone. That and, for terminally lazy cooks like myself, nectarines don't need to be peeled. (If you're not totally clear on how to peel a peach, watch Noelle Carter do it.)
When I'm at the market, I usually just pick up whichever is best that weekend. Whether it's peach or nectarine, I know I'm going to be happy.
How to choose: Select peaches and nectarines the same way. The first tip-off is color — not the red blush, but the background color, which should be golden. Really great fruit, which has hung on the tree long enough for full maturity, will even have an orange-ish cast to it. Great peaches and nectarines have an almost overwhelming perfume, too. Finally, ripe fruit will have a slight give when squeezed gently in the palm (gently!).
How to store: It's crucial to remember that peaches and nectarines will continue to ripen after they've been picked. If you choose fruit that is slightly firm (and, considering the way ripe fruit gets manhandled, maybe you should), just leave it on the counter at room temperature for a day or two and it will soften and ripen. Do not refrigerate peaches or nectarines until they are fully ripe — if you've ever bitten into a great-looking peach that turned out to be mealy and tasteless (and who hasn't?), it was almost certainly one that had been refrigerated before it was ready.