羡慕 means envy
嫉妒 means jealous, jealousy, this is normally a negative word.
here are some examples,
我羡慕你。 I envy you(maybe because you are good at singing.), which means that I hope to have the same thing that you have(I want to be as good at singing as you are.).
我嫉妒你。 I'm jealous of you(maybe because you are good at singing), the meaning overlaps with that of the previous sentence, but this sentence implies the probability that I might do something to stop you being that good, due to the fact that you are so good at singing, but I'm not as good as you.
I hope I've made myself clear.