Depends on the individuals and egos....DOP's usually worry about the lighting and have camera operators actually operate the camera and have very little to contribute to the framing, while Cinematographers light, frame and shoot. I just learned that 2 months ago on a shoot I was on. I never knew the difference, all these years...I thought it was just a different name for the same position. I worked with a guy on my film who was good at lightning but was clueless about framing....Some directors are better with actors than staging action, while some are better at framing action stuff than handling actors, rarely do you find someone who can do both equally. Ridely Scott would rather operate his own camera, like he did in the UK but the strong US unions get all pants in a bun about it.