There's a difference between cartoonish approach of SC1 and SC2.
The original had cartoonish elements and pop-culture references (from the Simpsons you mention, to the obvious Aliens and Starship Troopers). However, most of those were just flavour, in case you decide to click on a unit multiple times (something I always appreciated about Blizz games) - when it come to campaign-relevant, up-front goofiness, it was mostly reserved for Terrans, usually Marines (which somewhat serves to break the mold of WH40K Space Marines on which they were based), SCVs (stereotypical blue-collar redneck) and tanks, and again, it was there as a sort of flavor.
Other than that SC1 was a pretty serious, dark and violent game. Most Terrans, while not very deep, still rarely fall into the typical black-or-white character division (even Raynor, who is closest to a typical goody-two-shoes is an ex-criminal) - each having their own motivation and goal, and are often ruthless to achieve those - Mengsk, namely, who is intended to be a bad guy because of the stuff he has done, but it is shown that his regime is an improvement over the old government - or so I recall.
Protoss are shown to be the ones who bring order to Koprulu - a sort police of the galaxy - and a very brutal one. Noble as they are, they do not care if they commit a genocide of an entire planet due to a handful of Zerg being present on it, and they never show remorse for it (except for Tassadar, which is a major plot point). Other than that, Protoss are essentially as xenophobic as possible and don't care much about lives of those which they deem below them - like Eldar on which they were based.
Zerg, based on Tyranids, are beasts of war and carnage which survive on killing and assimilating other species. There is no morality to them, no remorse, nothing. They are pure engines of destruction. Kerrigan, who was a assassin-revolutionary with a murky past to begin with, fell from grace and became a "bitch-queen of universe", no less a monster than other zerg - just with human intelligence and cruelty. She is essentially evil incarnate - genocidal, cold, ruthless, manipulative and brutal.
So, in short, while SC1 does have a layer of cartoonishness to it, it is in its core a dark game. Not WH40K grim-dark, but still, there are very few positive things to find in this and its plot game, at least when it comes to the three main races. Most of the characters also have their bad sides shown, so there's that too. What's most important - the basic premise of the game is the struggle for power, survival and dominion in Koprulu - and absolute annihilation of anybody who opposes that.
SC2 took a full shit on all that.
Kerrigan is suddenly a good girl again, with a horrible past she can't account for. Terrans are full-on parody of what they were - the only logical thing to them is the fact that Raynor became an alcoholic. Protoss...well, we haven't seen much of them so far, but since they are working with Kerrigan and Terran against this bad ass Saturday morning cartoon evil guy who wants to destroy the universe™, I have no hope for them to be anything else but a joke of the original Protoss from SC1.
StarCraft was never a pinnacle of writing in video games, and had quite a few crimes to account for (UED for example), but SC2 is a whole level below it. From a relatively "adult" SF video game, it went to a cartoon for kids with an incredibly simplistic good against evil plot, cliche and flat characters. In fact, I'd say that the decline of SC's storyline started with some elements of Brood War (the mentioned UED and hybrids) - or at least, the shit that SC2 is evolved from BW's lowest points.