The sentence goes one of these two ways:
"He knows WHAT other teachers don't know"
"He knows things THAT others teachers don't"
Which is better to use - WHAT or THAT? Are they both grammatically acceptable?
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Both are grammatical, but they mean somewhat different things.
He knows what other teachers don't know.
In this the clause what other teachers don't know is a "free relative clause", which acts all-by-itself as a noun clause. In this case it acts as the Direct Object of the verb knows.
He knows [DirObj what other teachers don't know.]
In another sentence it might act as the Subject:
[Subj What other teachers don't know] is how to communicate with their students.
If it is not specifically modified, it will be understood to mean everything which other teachers don't know.
He knows all the things which other teachers don't know.
He knows things that other teachers don't know.
In this the clause that other teachers don't know is a "bound relative clause". It might also be written which other teachers don't know. This cannot stand alone: it has to follow a noun phrase (in this case things) which it modifies. In effect, it acts as an "adjective". As FumbleFingers points out, if you said the things here it would be equivalent to your first sentence; but the version without the the says that there are merely some things which he knows but other teachers don't:
He knows some things which other teachers don't know.