Let's use the aftermath of Superstorm Sandy as an example to illustrate the differences between collision and comprehensive. Within that storm, let's consider two events that might have happened: 1) a heavy tree branch fell on your car, or 2) you swerved to avoid a falling tree branch and wound up crashing into a tree. In the first event, you had no control over when or why a tree branch would fall on your car. This kind of accident would get reimbursed under your comprehensive policy. In the second situation, you were driving the car and ultimately swerved into the tree, which makes it a collision, and collision insurance therefore pays for the damages. Events like the hypothetical ones stated above are why it's important to differentiate between the two types of coverage.
In the states with no-fault insurance, insured drivers are typically paid for medical expenses by their own insurers, regardless of who caused the accident. Nonetheless, BI liability coverage is still required in no-fault states because if injuries are bad, the at-fault driver may be sued by the injured party. If that happens, your BI coverage can help cover your liability expenses.