Inspire is a transitive verb. (X inspires Y)
Inspire means; to make someone have a particular strong feeling; to make someone feel that they want to do something and can do it.
Aspire is an intransitive verb. (X aspires.)
Aspire means; to want something very much; to hope to achieve something or to be successful.
We can inspire someone. When we inspire someone we fill them with the urge to do something or to feel something. So, inspire is about something that is not within us, but external. (external effect).
However, aspire means to hope, to have a strong desire to achieve something. Thus, aspire is about internal effect, it is about something within us.