worry is a verb. You worry about someone or something. Anxiety is a noun. I feel a lot of anxiety about taking the math test.

You would worry about a family member who is sick. (Can't use anxiety ) Anxiety: 'He caused his parents great anxiety by cycling great distances alone' (Can't use worry) 'We waited for news with a growing sense of anxiety. (Can't use worry)

None really although if somebody worries a lot they may be given a diagnosis of suffering from an anxiety state.