Originally posted by Tyrannosauruschex
I think the position dictates heavily which piece is more useful but here are some points.
1. The longer the game goes, and the less pieces left on the board - the less restricted the knights become, so are moe useful
2. Knights are the only piece that can attack any other piece without fear of being themselves attacked.
3.Fianchettoed bis ...[text shortened]... knight but, as with all endgames, king placement will decide the game more so then either piece.
Well you are clearly better than me, so these guidelines obviously serve you well and I am cautious in challenging them, but whilst points 2 and 3 are undeniably true, I would offer an alternative view to points 1 and 4.
For point 1, I would say fewer pieces and hence more space often favours the bishop over the knight. The former is more mobile, and this is enhanced on an open board. Knights gain an advantage in more closed positions simply by being able to jump! Of course knights can control squares of either colour, but in many types of endgames they lack the control of multiple areas of the board, and cannot relocate as quickly as bishops.
For point 4, my understanding (as per the points above) was that bishops were a match for knights in the endgame and often (such as two bishops against any other combination) have a distinct edge. Of course, king placement is key here as you point out.