First of all, a criminal case is different from a civil case. In some situations, a defendant’s act is both a crime and a civil wrong, called a “tort.” In other situations, it might only be a tort.
The prosecutor is responsible for enforcing criminal laws. A civil wrongful death lawsuit must be enforced by the loved one’s surviving family members. Both cases can be going on at the same time.
For another reason, some conduct that is a civil wrong is not a crime. For example, a person who carelessly backed out of their driveway without checking their rearview mirror has been negligent but probably not criminally negligent. This means they can be held responsible for a wrongful death but probably not for murder or manslaughter.