DKC2 pretty much improved upon DKC in every single area.
From a level design standpoint DKC2 walks the line between DKC1 and DKC3, DKC1 has pretty basic level design, lots of running forward and bouncing off tires, there's a few interesting gimmicks in there but it felt like it was still trying to find its footing and its much praised underwater levels were often incredibly bland. DKC3 meanwhile is gimmicks galore with no filter for what was a good idea or a bad idea. So DKC2 being in the middle here gave it a stronger emphasis on interesting gimmick stages alongside some more traditional platforming stages, outside of Red Hot Ride which is far too slow for my liking all its gimmicks tended to work well and each stage has something unique to make them stand out right down to the water stages as well.
Animal buddies play a bigger role and the newcomers are fleshed out enough that they can hold entire levels to themselves.
The aesthetics of the game stand out a lot more, it's not just a case of things being more moodier under the grimmer direction of Crocodile Isle, there's the fact that stage archetypes didn't limit themselves so much to the more grounded look of it predecessor and sequel. You'd ride ramshackle rollercoasters, clamber up the sticky honey walls of a zinger hive and journey through twisting bramble mazes up in the clouds, it's all more fantastical and memorable. Even when revisiting an old stage concept in the jungle theme it gets a dense tribal makeover to more sufficiently fit the games bleaker tone.
Even the enemies got retooled under this direction, Pirate Kremlings made for more visually intersting adversaries and led to some more memorable enemies, Krook, Kloak and Kannon do great jobs with the whole pirate gimmick, even the Neckies got in the act sporting a spiffy bandanna. K.Rool himself goes from fat croc that has a generic attack pattern of jumping around, crown throwing and dashing at the player to Kaptain K.Rool with a blunderbuss packing more functions than a swiss army knife making for a far grander finale, bosses on the whole are a big step up but it's hard not be coming off Boss Dumb Drum and Really Gnawty Beaver.
Secret placement in the first DKC is pretty shocking at times, while Rare would leave some areas hinted at by the odd banana other times were just shots in the dark hinted by nothing at all, a game of throwing yourself into pits in the pursuit of bonus rooms which didn't lead to anything particularly interesting. DKC2 placed a greater emphasis on secrets, it hinted towards them more and had Cranky on hand to throw a few hints your way, on top of this the bonus rooms led you to extra levels giving a greater incentive to seek them out. also worth pointing out that DKC2's change to bonus rooms as more like challenge rooms is one of the game's more underrated changes for the better, taking either the stage gimmick or element of the game in general to deliver something unique no matter the limited tasks of collecting stars or bashing baddies.
The duo dynamic is more improved, the Kongs being able to throw each other into tougher enemies or up to higher places feels more teamwork focused than say switching to Donkey in order to jump on a Krusha because Diddy can't. Despite both being characters of a similar build Dixie differentiates herself with the helicopter spin while Diddy gets to be touch quicker with a slightly higher jump. Yeah i've seen a few times that people say the first DKC had it better for the whole DK weight factor over Diddy's sort of speed, in that case DKC3 wins all because it has all of the above and then some.
And then there's the music, Dave Wise knocked it into orbit, what more needs to be said?
Overall there's more polish be it better (if still not perfect) collision detection, less cheap shots, a better defined difficulty curve and more varied stage styles with certain archetypes focusing entirely on vertical progression.
Not bad for a game smashed out a mere year later, now for the love of all that is decent Nintendo get this on the Wii U VC or get those 3D Genesis/Mega Drive fellas to work their arcane magic on a 3D DKC trilogy.