In most common settings, we assure someone to put their mind at ease so they won't worry. We insure a person or thing by buying an insurance policy on them that will pay us money about their death/destruction.
He assured his mother that he wouldn't do anything dangerous.
His mother didn't feel assured (didn't believe him) and insured him with a $1 million insurance policy.
He was not assured by his mother's lack of confidence in his ability to stay out of danger.