How does "among" vs "between" affect the meaning of this sentence from the Times?

It strictly is neither intra- nor inter-group, as the context shows.

As you seem to know already, it is between two objects, and among more than two.

The differences referred to are between (each individual member of one group) and (each member of the other) -- a many-to-many relation, so we need among here.

intra: The juniors are quiet, but the seniors fight among themselves.
inter: There will be a friendly match between the seniors and the juniors.
Our case: There are friends among the juniors and the seniors.
inter, 1-to-1: There will be boxing contests between the juniors and the seniors.

Many writers insist that "between" should be used only when you're talking about two things or two people, etc. and that we use "among" when we're talking about more than two. However, "between" is actually quite useful and acceptable when describing specific differences that exist both individually and severally between one thing and several others. "Among," according to the OED, expresses differences that exist in a vague and collective way. So a lot depends on what you mean to say, but "between" might be acceptable in the sentence above if it was referring to specific differences.

Authority: The New Fowler's Modern English Usage edited by R.W. Burchfield. Clarendon Press: Oxford, England. 1996.

In the context of the article, the inequality is between the rich group and the middle-class group. I would have expressed it with between rather than among, but I acknowledge @Robusto's sagacity: "Rule of thumb: If you see it in The New York Times and it's not an obviouss typo, you should probably assume it's been copy-edited by people who are a lot fussier than you are."

I think I have to go out on a limb here and say this is poor phrasing by NYT. Having read the article, it seems clear to me that the two inequalities being referred to are between the middle class and the bulk of the populace, and between the rich and the bulk of the populace.

Using among suggests that inequality has increased within each of those two groups, considered in isolation, which is not consistent with anything else in the article (or indeed common sense).

If it had been between, that would mean the difference between the wealth of the middle classes relative to the rich has increased. That's at least a plausible observation to make, but again, it's not consistent with anything else in the article.

I don't actually think there is a truly correct short form of words to convey the intended meaning, but personally I would go for...

inequality by the middle class and the rich.

(i.e. - caused by the middle class and the rich taking a disproportionate share of the wealth)