I think the big difference is that a class III if done correctly identifies high risk medical problems for pilot incapacitation, or degraded performance. Some examples include but are not limited to insulin requiring diabetes, seizure disorders, heart failure, coronary artery disease, suicidal depression, severe neuromuscular disease, disqualifying medications like prescription narcotics etc. You can have all those conditions and if your doc says you are stable, he can sign off on your DL, and can sign off on your basic med if he/eh feels you are OK, especially if they have no real knowledge of the demands of flying and flight physiology. Depending on case, those medical problems might not be compatible with the pilot that you want flying your family. The fact that flying is as safe as it is from a medical incapacitation standpoint is as much about the people that are never allowed to be pilots, as it is people that are pilots that are heavily monitored.