In short, a show doesn't have the possibility of being a decent story if it doesn't have a consistent narrative.
Not all shows with a consistent narrative have a good story, but all shows with a good story have a consistent narrative.
That's not to suggest that "good story" is the be-all and end-all of a good show, but it's what we're discussing at the moment. That aside, I tend to enjoy things based on how much they appeal to me, not because of their association with a certain style or certain tropes. For me, good story is a large draw; anime series by default are more likely to have a consistent narrative (and thus potentially a good story) than western animation. I still enjoy western animation, but for (somewhat) different reasons.
In short, a more apt comparison would be between anime and live-action series in the West; both are story- and character-driven. So, for example, a comparison between Cowboy Bebop and Firefly would be much more apt than one between the former and Spongebob, or between the latter and, say, one of the Touhou games. The 'animated' aspect of things is largely irrelevant, except in terms of breaking down things into tropes and certain stylistic choices which become available.
Re: Western "adult" animation: For the most part, it's as described: dirty humor. Some of it is good, some of it is crap (Family Guy). There's plenty of potential for more serious western animation, but it'd have to overcome the cultural bias against the same.