We often use those two words interchangeably, but the first means "I want that, too" and the second means "He shouldn't have that."
Envy is a subset of greed, but it can have a productive outlet. The kid who wants a bike like the kid next door might mow lawns or babysit to get the money to buy one himself. The envy led to something positive.
Jealousy can have greed attached, but is more destructive. The jealous kid can steal the bike and hide it from his neighbor, or slash the tires.
Envying the rich can lead one to strive to get their oneself. I recall mentioning to a finance student the wealth that investment bankers ring up; the student's reply was to want to get into that line of work themselves.
Jealousy of the rich can lead to theft and political class warfare. If we can't keep up with the Joneses, we bring them down to our level, even if it doesn't help us much.
The class warfare goes both ways; I saw a piece the other day of a pastry-shop owner not wanting to take electronic food stamps, since folks on welfare didn't need stuff that nice on the public dime. Do the poor not deserve nice things from time to time? Yes, there is more than a bit of food stamp abuse, but there is a jealousy there ("they don't deserve that, not with our money") that isn't pretty.
None of us sinful critters deserve a good Boston creme pie, but some of us have enough disposable income to afford it; even a poor person on the dole might have enough above mere subsistence cash to afford a few frills and not be seen as a drain on society for it.
Jealousy isn't pretty. Envy (or covetousness, if we want to go old-school) isn't pretty, either, but it's not quite as ugly in the long haul.