Well said Polycrates - you put words to the vague ideas I have in support of the Inca. Plus, they're fun to play! No other DLC civ have I played as often, and I think they're just out and out one of the most fun civs to play.
The extra mobility in hills really seems to open up the land, as do the extra free roads Inca's can have. The extra free gold from trade route maintenance makes a big difference in large empires, and more gold always gives more options - the flexibility of Inca is probably a big part of why it feels fun to me.
Songhai is another civ that captures that amorphous personal sense of fun for me - barb hunting, and getting big chunks of extra gold is fun. It makes the game more engaging.
Babylon is probably the strongest civ in the game. But it's far from the most fun - for me, at least.