I don't think either mode necessarily presents more potential harm; that said, most people performing bodyweight-only movements are doing so at a basic or intermediate level, i.e. not performing movements with significantly increased moments such as planches, levers, etc. On the other hand, when lifting weights it's relatively easy to throw around large loads, even loads greater than technique and strength would theoretically allow. That being the case, I can see weightlifting being a more common source of pain and/or injury.
In specific regards to shoulder pain, it nearly invariably seems to arise from even the most minute deviations from proper technique. For example, allowing the shoulders to collapse when lifting overhead or receving a weight overhead as in a jerk.
Gymnasts demonstrate the possibility of becoming very strong using entirely or primarily bodyweight only movement. Achieving extremely high degrees of strength and power this way probably requires a lot more high-end coaching and intelligent programming, however. I would also think leg and hip strength and power would be much more difficult to develop to the same degree as those traits in the upper body without external resistance.
Bottom line, I personally wouldn't choose one over the other--I would suggest devising some programming that incorporates the best of both modes and avoids movements that you know cause you pain. But before eliminating anything, I would be very sure any problems are not solvable through technique corrections or some injury treatment.