Even if you think they’re dying, newspapers have something your Twitter stream doesn’t: hierarchy of what’s important to read.
Story “weight” is intuitive on paper. There’s what’s above-the-fold, and on top of that, there’s clear positioning of pieces, with one more prominent than another. There’s differences in headline size, perhaps subheads. In some cases, there’s teasers to other stories to read once you’re inside the paper. When you get to the actual stories themselves, often times there’s another indicator: length in inches. Design works to show your eyes where to go, and what is editorially important to look over (perhaps over cereal, or a cup of coffee).
Home pages replicate this idea in part. Article pages are getting better at this, or at least people are making a case for it. Apps for tablets often do this as digitally close to a newspaper as possible. But social media doesn’t really replicate the “story weight” capability of a paper at all.