If you are at fault for the accident your insurance premium will increase, and even if the accident's not your fault your premium could increase. The determining factors include, your driving record, claims history, and type of insurance policy you have.
Driving without a license is a violation of the law, but it's also a completely separate issue from your claim for damages. The party at fault is responsible for the damages to your vehicle and any injuries you have sustained.
Victims of car accidents can have delayed injury-related pain. They may have even suffered a severe injury and not realized it at the time of the collision.