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.
Insurers don't determine your actual cash value (ACV) settlement based on what you owe, but rather on what the car is worth just prior to the accident. Let's say you owe $20,000 on your new car, but it's only worth about $16,000. If your car is totaled, you might get a settlement check of $16,000 but still owe an additional $4,000 on your loan or lease.