The term ipse dixit is the Latin equivalent of the Greek autos epha,
referring to Pythagoras, as in, The master (Pythagoras) said it so it must
be true and no proof is needed. In our modern world, this has many forms:

Child: Why do I have to go to bed at eight every day?
Parent: Because I said so.

Employee: Why do we have to do this project if it's going to be scrapped anyway?
Boss: Because I said so.

An appeal to authority also known as argument from authority, argumentum ad verecundiam (Latin: argument from modesty) or ipse dixit (Latin: he himself, said it), is one method of obtaining propositional knowledge.

What's the difference in the usage of ipse dixit and quod erat demonstratum?

Ipse dixit would be said after a successful Proof by Blatant Assertion, quod erat demonstrandum after a real proof.

Yup. "Because he says so" ain't QUITE "this is proven".

Actually quod erat demonstrandum doesn't mean 'this is proven', it means 'which was to be proved", i.e., this is what we had to prove. It implies 'and now we have proved it' but it doesn't mean that.

Bingley

indeed, quod erat demonstrandum is usually said/written immediately after the last sentence in a proof, in which we have finished to prove what we had to prove

I know, I know. But the "truth" didn't fit what I wanted to say exactly, so I newsmediaed it.