Nectarines are actually a type of peach without the fuzz. They are the same genetically, and just one recessive gene is responsible for the fuzzless skin.
But that doesn't mean they taste the same (as you've probably noticed). Nectarines definitely have similar flavor to peaches, but distinct differences too. Nectarines tend to be smaller, firmer, and have more honey-like aromas.
Soft nectarines should be stored in the fridge if they aren't being eaten right away. Their quality will remain good for a few days, after which their flavor and texture might deteriorate.
Hard nectarines can be left on the counter, preferably out of direct sunlight. They will usually soften enough to eat in 1-3 days. Once soft, you can store them in the fridge if you're not ready to eat them.