When you're using the HDMI cable, the monitor can tell the player what its resolution is. The recorder then takes the 720xwhatever DVD resolution, and rescales it to the resolution of the monitor. The logic is that doing upscaling in the player, the player may do a better job of upscaling than letting the monitor do it. In reality, some players do a better job, and some do a worse job.
In both cases, there's no magical extra detail.
Your recoder, incidentally, can record onto DVD+R, DVD-R and DVD+R DL (that's the dual layer you were wondering about). In general you'll get a better recording on +R and on +R DL than on -R.
In terms of video quality, on your machine it's HDMI, component and S-Video in descending order.