Best thing about bivving for me are the stars. I just look up, usually see loads of shooting stars, try to recognise as many constellations as I can, and since I'm always with friends we usually speek till we fall asleep. I also hate leaving the atmosphere of the camp fire, friends and all to go to shut myself up in a tent alone or with just one other person, with no view at all...I prefer keeping that atmosphere till I fall asleep (but then it's by watching the stars, no the fire).
So best thing for me are the stars, but with of course everything that goes round them (place, noises etc.). Actually, the french expression for sleeping with no tent or tarp or anything is "dormir à la belle étoile". It would translate to a less poetic "sleep at the pretty star", I think it suits the activity just right :-D
P.S: Young bushman: take all your normal kit (tent and all) and if it's a clear sky and nice weather just sleep out of the tent. Or if your absolutely sure it will be nice weather (is that even possible in britain?
): ) just don't take your tent at all.