Assurance is some form of a guarantee and insurance is a guarantee of compensation.

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.