Diddy cartwheels, not that much changes in regards to that. Also, the Peanut Popguns have been Diddy's staple since DK64, as are his Barrel Jetpacks. The Barrel Jetpacks are entirely exclusive to Diddy.
EDIT: Naturally, though, the peanut popguns are virtually the same as Dixie's Gumball Popgun (which debuted in Tropical Freeze) and DK's Coconut Gun (which did not return, though it may yet if they give DK the abilities of the other Kongs; shooting and a second jump).
In DK64, Diddy had two abilities that haven't appeared since: Diddy had a very high jump where he coiled up his tail to jump higher, and a ram attack where he rams his head into things which can be used to activate switches and ring gongs. Neither of which he uses in Smash as far as I'm aware.
Diddy's main trait is being the fastest of the bunch. Dixie is semi-fast while DK is rather slow, but Diddy is always the fastest Kong. Except in DK64 where Lanky has some moves which make him faster, though I think Diddy's default running speed is still faster than Lanky.