The exchange itself is funny and more than a bit silly but there are some serious things to take away from it.   Robin has two posts describing and summarizing the exchange:

• Yesterday, I got into an argument with Chelsea Clinton. On Twitter. About Hannah Arendt.
• Chelsea and Me: On the politics—or non-politics or pseudo-politics—of engaging a power player on Twitter

Robin’s first post describes the exchange it.  Unless you’re actually interested in the details, you can skip it.   The significant takeaway from my perspective is in his second post (emphasis mine):

…Had this exchange occurred with a Republican, or with one of the sons or daughters of a famous Republican, say Eric or Ivanka Trump, I have no doubt that I’d be hearing nothing but lusty cheers and congratulations, particularly from Democratic Party partisans. I mean these are folks who manage to muster a fresh cackle at every prodigy of stupidity the right manages to produce on any given day. But Chelsea Clinton is part of the team, so, well, the obvious. And that’s fine; I don’t begrudge people their partisanship. But I do ask that they cop to it and not pretend that I’ve somehow transgressed a norm they’d never acknowledge if the other party were on the receiving end of it.

Second, there’s a related element that’s worth noting. And that has to do with the politics of intelligence/education, social class, and partisanship. The Democratic Party and its supporters like to think of themselves as the party of the smarties. Obama, Clinton, Clinton, Clinton: all so smart, all so well educated, all so well spoken. That’s why they’re entitled to rule, their supporters think. (Believe me, I’ve had these conversations many times.) And that’s not just about politics; it’s also about social class, or at least the culture and style and markers of a particular kind of social class. Unlike the Trumps and other vulgarians of the right, these are people who know how to carry on a conversation at a cocktail party or on Charlie Rose. (Is that show still on, by the way?) Indeed, a well educated liberal person on Twitter—a professor of political science, in fact—made a point of noting to me that none of the Trump kids had read Hannah Arendt. That Clinton didn’t seem to get Arendt didn’t matter. It was enough that she had read Arendt. Or knew to show that she had read Arendt.

I’ll confess, I find that kind of thing distasteful. (Arendt has a great line in Eichmann about how the well-heeled educated German classes of the postwar era didn’t really have a problem with the fact that the workaday Jews of Germany—little Hans Cohn from around the corner was how, I think, she put it—had been murdered during the Holocaust; it was that Einstein had been sent packing. That was the real crime: the loss of all that wondrous cultural capital.)

My objection is not just academic or aesthetic or cultural; it’s also political. I don’t believe in technocracy. I don’t think I (or people like me) am qualified to lead the country or to have a Clinton-like position in this country because I went to good schools or read a lot of books. There’s a limited place for expertise in a democracy, but it’s limited. I know I’m in the minority here on this, but I get no comfort from the fact that Barack Obama reads great literature (that was a Facebook post a while back) or that Chelsea Clinton knows how to name drop Arendt. For me, that doesn’t reflect the legitimate needs for some limited expertise. Nor does it reflect the requirements of good leadership, and it sure as shit is not about democracy. It’s about social class, social standing, and social signaling.

See also Steve Randy Waldman on the limits of technocracy, Your Theory of Politics Is Wrong.

We Democrats can be an anti-authoritarian party or a party of the benevolent aristocracy.  We cannot be both.