In reference to recumbent type bicycles recumbents have a strait back or school chair like seating while supine are far more laid back more like a recliner seating position.
these are some examples I found for supine
http://www.flyingfurniture.com.au/m5.html
or
http://www.catbike.us/
^these are more laid back
& those below are more of a chair seat.
THese two links are more standard recumbents
http://www.easyracer.com/
or
http://www.lightningbikes.com/
I prefer mine supine because my weight is spread out more over the seat while with standard recumbent more of you weight is resting on your behind and less on your back.
As I thought about this and saw the word play I realize that mine is not the answer you want.
Both recumbent and supine are laying down postures the difference in reference to word play is that supine the head is forward lying face upward and in recumbent it is laying down. Or as best I can figure supine is like a sleeping position and recumbent is more like a relaxed awake reclining position. < of course that is reference to people again and the words go farther than just people.
As I see it supine is flat or horizontal and recumbent is slightly inclined but basically horizontal