ようとしない

ようとしない means "won't", "refuses to": someone could do it and deliberately doesn't, and the speaker is usually frustrated by it. The verb goes in the volitional form, followed by としない, the negative of とする.

The verb before と is the volitional, not the dictionary form: ×食べるとしない.

ようとしない or ない

A plain negative states a fact. ようとしない adds that the person is unwilling.

About other people

It describes someone else's unwillingness as you see it from outside, so the subject is almost always another person or an animal. For yourself, say what you don't want to do with たくない.