Strike up another one for Yoshi's Island. I agree that the level design in that game is better, but I also like its gameplay more. DKC's controls are more basic, and aside from some pretty awesome rolling jumps and barrel throwing, there's not really much else you can do aside from the basic moveset. Fitting with Yoshi Island's more expansive level design, Yoshi feels like a much more maneuverable character, and there's a lot of ways that you can deal with obstacles and enemies, all of which are easy to learn and can be employed to great effect with tricky platforming and puzzles.
Aesthetically, I think Yoshi's Island looks both more enjoyable and impressive, but that's primarily because I personally like more stylized artwork in general. The one thing I will say that the DKC series has over Yoshi's Island is a better overall soundtrack, which is primarily due to the songs in both games being of a similar quality and DKC having a greater variety. The Athletic Theme is great and whimsical and really fits the tenor of Yoshi's Island, but as one of about 4 distinct level themes, it's going to get stale fast. The awesome gameplay trumps this about 98% of the time, but it's still the biggest complaint I have about the game.