On the topic of DK1 atmosphere v DK2 atmosphere, I would argue simultaneously that 1) this forum's preteen attitude toward what makes something mature (i.e., seriousness and darks) is clouding the discussion, and 2) DK2 was definitely lighter, in a number of ways. As some have mentioned, the goofiness of DK2 was primarily in the cinematics, of which there were tons, but there's no way that shiny goofball Lord of the Land would show up in DK1.
DK1 was funny as all hell, but much more subtle and dark in a fashionable '90s gothy Rob Zombie Diablo 1 kind of way. I remember seeing a prerelease magazine ad for it that literally just had an actor dressed as a medieval executioner and holding a red-hot iron. It was fucking creepy. The ad probably said something like, "It's good to be evil."
However, gameplay-wise, the difference in vibes I'd attribute mostly to graphics technology, not to the style of game. The resolution and engine of DK1 make it feel grittier and bulkier. DK2 was very shiny, crisp, airy. Everything felt so thin, just due to the models. DK1 monsters feel heavy. They're just blurry blobs and they feel thick. You feel the mass of bile demons and demon spawn. DK2 removed demon spawn, which is something I never really approved of, and it replaced them with a bunch of humanoids like rogues, goblins, dark elves, dark angels -- basically, thin things. It also got rid of the whole get-bigger-as-you-train thing that I loved, and also salamanders didn't turn into dragons at level 11 -- indeed, they fuckin' removed dragons.
So basically, I think it's an 1999 art engine thing combined with the fact that they replaced bulky things like demon spawn and beetles with thin humanoids.