The closest answer I can come to is this:
"If it [the Ring] is destroyed, then he will fall, and his fall will be so low that none can foresee his arising ever again. For he will lose the best part of the strength that was native to him in his beginning, and all that was made or begun with that power will crumble, and he will be maimed for ever, becoming a mere spirit of malice that gnaws itself in the shadows, but cannot again grow or take shape. And so a great evil of this world will be removed."THE RETURN OF THE KING, "THE LAST DEBATE"
The last sentence is pretty straightforward. Sauron's eternal punishment was to forever wander the earth with no power to do anything.
My understanding is that, like the game they were playing on who got the most kills, it was a dwarves vs elves type of challenge, who was the better fighter, soldier, etc. Gimli and Legolas started off with a lot of shared animosity, due to past grievances between their two races. At this point, though, a friendship was developing, so it was a friendly challenge, but to have to be tossed by Aragorn would make Gimli look like the lesser soldier, because he knew Legolas could have done it alone. All about saving face.