In general, it’s always more convenient to deal with your own insurance company after a car accident—but there are some specific instances where you’ll probably want (or need) to file a claim directly with the other driver’s insurance company instead.
If you only have liability insurance coverage on your vehicle, your only real option is to file a claim with the other driver’s insurance company. Similarly, if your accident was relatively minor in terms of damages (i.e., less than the cost of your deductible) and fault is not being contested, it may make financial sense to file directly with the other driver’s insurance company.
One benefit of filing a claim with the other driver’s insurance company is that there’s no deducible to pay. The bad part is that the claims process can move very slowly, and there’s no guarantee they’ll even accept liability. Their interests are completely counter to yours, and they simply have no incentive to move quickly or negotiate fairly (unless you have an attorney).
In almost every minor motor vehicle accident (i.e., property-damage only; no injuries), it’s generally in your best interests to file a claim with your own insurance company. It’s part of what you pay them for. While you will have to pay a deductible up front, your vehicle will almost-certainly get fixed quicker. Best of all, if the other driver is found to be at-fault, your insurance company will eventually reimburse you for your deductible.