Don't confuse things. It is not a matter of PPL or CPL, it is a matter of skills. I have had many ATPL guys taking a checkride and performing uttermost bad in a small aircraft. Being a commercial pilot and running under AOC won't necessarily qualify to take passengers in a PA28! In fact, not few ATPL riders have less hours actually on the yoke than a typical PPL pilot, quite some fly by buttons on a glass cockpit only. This may be one reason for the confusion outside the pilots community, many license arguments are insurance discussion only.