Usually, the other party’s insurance company will advise you to use your own insurance or to pay out of pocket for your own damages only under certain circumstances. These circumstances could involve a question of liability or a question of coverage.
If the other party described a different version of the events causing the accident, for example, the other company might still be investigating liability and would not yet commit to paying your claim.
If there is a question of coverage, the company is attempting to verify whether the other party had valid coverage at the time of the accident, and if so, what coverages would apply.
In either case, obtaining a copy of the police report may help to resolve your claim.
If you have collision coverage on your policy and elect to use your own coverage, then your insurance company would go after the other party or their insurer to recover anything paid on the claim, as well as your deductible, in a process called subrogation.