Translation:He does not speak Chinese.
It's a tempting distinction with some validity, but it's not strictly adhered to in practice.
The terms are "汉语" and "中文" are generally equivalent, though apparently subject to regional preference, as set out in this Chinese Wikipedia article:
In China, both terms are fairly common, in my experience. In Taiwan, "汉语" is rare.
The terms don't overlap completely in their usage (e.g. "中文" rather than "汉语" would be a course of post-secondary study for native Chinese speakers equivalent to "English" in the English-speaking world), but generally either can be used to describe the spoken language (and by extension, its transcription in writing).
It's not really accurate for "Chinese", which is a broader term. "普通话" specifically means Mandarin.
Even more specifically, it's the mainland term for Modern Standard (Mandarin) Chinese (called "国语" in Taiwan and "华语" in Singapore and Malaysia). We usually simply call this "Mandarin" in English, but Modern Standard Mandarin Chinese is derived from 官话, the language of the mandarins (senior-level Chinese bureaucrats), which is the historical technical meaning of "Mandarin".
(Mandarin as the language of the officials was in turn based on dialects existing in and around Beijing, which are still part of the so-called Mandarin language group as it's described by linguists.)
Thank you. This skill has confused me because I was taught never to say "Chinese" because the different dialects are basically completely separate languages. So it seems that what Duolingo translates as "Chinese" can refer to any dialect spoken by the Han Chinese, which in the context of this course probably means the dialect we are learning. Which Chinese people are Han?
Most Chinese citizens are Han — 92% according to official Chinese statistics — and accordingly so are most Chinese emigrants. The Han are who we generally think of when we think of Chinese ethnicity, I would say.
There are also ethnic minorities in/from China (i.e. current or former Chinese citizens who aren't Han), and although many of them have largely mixed with the Han people, some groups still speak non-Sinitic languages.
With the 会, that would mean "He does not know how to speak Chinese" or "he does not have the capacity to speak Chinese;" the sentence as written means "He does not speak Chinese;" perhaps the former is the reason for the latter, but the sentence does not actually specify: that is, it is possible that he can speak Chinese, but he simply does not, for whatever reason.
他不说汉语。 Translation: He does not speak Chinese.