When we say we admire someone or something, we are usually thinking about their character or essential qualities. If I admire this person today, I will probably still admire them tomorrow, because I admire them for the type of person they are.
When we say we appreciate something, we are usually thinking of a particular thing that has occurred or has been done. We do sometimes say that we appreciate a person, but even then we are usually meaning that we appreciate their contribution to something, or something that they have done.