Homesickness is a negative feeling you get when you don't like the place where you are living currently and you wish you were back home.
"After a month at college, she was very homesick."
Nostalgia just means a pleasant, sad or longing feeling about a past event. You could be happy in the situation your are in and still feel nostalgia for a past time. It may or may not be connected with negative feelings.
"That old tune gave me a feeling of nostalgia for the 60's"