With background images you lose the ability to scale an image to fit its container, which is one of the key components of a truly adaptive (fluid) layout. However, if you are using a fixed grid that merely resizes at certain thresholds in viewport width (like bootstrap's adaptive grids), that will work fine.
Regarding mobile, I'm not sure the separation is about performance as much as it is about being nice to users with capped data plans. And even with a high resolution mobile display, you just don't need the same size images for a 3 inch phone as you do for a 24 inch desktop monitor. In fact, I think one of the intentions of the responsive design movement is to serve identical experiences to desktops and mobiles, while adjusting the format somewhat to make the latter easier on the eyes. This is a huge improvement over serving one of those dumbed-down, text only mobile sites that blogs and such are so fond of redirecting you to (I'm looking at you, wordpress).