Is there any difference in sound quality when recording from either one?
From a sound card, probably not. From a large amplifier or sound mixer, there might be. The headphone system has to go through a speaker amplifier and that can add slight distortion, but it only exhibits the distortion when it’s driving headphones or small speakers.
The controls for each are different, but beyond that not much. I would stick with the jobs that the connections were intended to do. It’s one less thing to worry about if you get into trouble.