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.
While there are a couple of benefits of collision insurance, the main one is that you can file a claim and receive reimbursement regardless of who was at fault. Collision claims usually get processed faster than property damage claims because the insurance company does not have to spend time investigating who was at-fault. Another benefit is that you only deal with your own insurance company, rather than another insurer with less incentive to pay for your claim. Collision insurance can also be used toward your rental car in most cases, which can spare you from having to buy rental car insurance.
Actual cash value equals the purchase price of your car minus depreciation and your deductible. So comprehensive coverage will pay an amount up to the actual cash value of your car to either repair or (in the case of a total loss) replace it. If the cost of repairs exceeds your car's ACV, your car insurance company will declare it a total loss and pay the sum of the car's ACV to help you replace it — unless you opt to retain salvage (i.e., keep the totaled car), in which case the salvage value will also be deducted from your payout.
When you purchase comprehensive coverage, you will select a set deductible, which is the amount you pay out of pocket toward a covered claim. Let's say you choose a $500 deductible, and your car is later damaged by hail in a covered claim. If it costs $1,500 to repair your car, you would pay your $500 deductible, and your insurance would pay the remaining $1,000.