"Homesickness" is usually used to describe the feeling of homesickness after leaving home, feeling uncomfortable in a strange environment and longing to return to a familiar hometown. Nostalgia, on the other hand, more broadly expresses the nostalgia for a certain period, a certain place or a certain person in the past.
Here are some examples:
When he moved to this city, he had a strong feeling of homesickness.
When he moved to this city, he was very homesick. )
Old photos of his hometown always evoke nostalgia.
Old photos of his hometown always make him miss. )
This novel profoundly depicts the hero's nostalgia for his rural growth experience.
This novel vividly describes the hero's nostalgia for his rural growth. )