The two uses originated separately, with the diaeresis being considerably older.
From the en.wikipedia.org
In French, in such cases the diaeresis is usually written over the following vowel.
From the en.wikipedia.org
This diaeresis is not used over a stressed vowel that already should have an accent.
From the en.wikipedia.org
In the few exceptions where the u is pronounced, a diaeresis is added to it.
From the en.wikipedia.org
Historically A-diaeresis was written as an A with two dots above the letter.
From the en.wikipedia.org
In poetry, the diaeresis may be used on i and u as a way to force a hiatus.
From the en.wikipedia.org
It typically serves as a pronunciation guide, e.g., to mark a diaeresis.
From the en.wikipedia.org
Sometimes, there's a need to distinguish between the umlaut sign and the diaeresis sign.
From the en.wikipedia.org
Modern Greek has used only two diacritics since 1982, namely the diaeresis and the acute.
From the en.wikipedia.org
More examples
Umlaut: a diacritical mark (two dots) placed over a vowel in German to indicate a change in sound
Diaeresis (dieresis, diu00EBresis) /dau026Au02C8u025Bru1D7Bsu1D7Bs/ may refer to:
An umlaut is the orthographical representation of a type of sound shift in spoken language. A very similar diacritical mark (called diaeresis or "trema") is used to signify a linguistic hiatus. ...
"u00C4" and "u00E4" are both characters that represent either a letter from several extended Latin alphabets, or the letter A with an umlaut mark or diaeresis.
Alternative spelling of diaeresis / dieresis
A pause in mid-line where the end of a word matches the end of a metrical foot. In elegaic couplet, it usually occurs in the center of a pentameter. It functions much the same as caesura. ...
Two " placed site-by-side over a vowel (like an umlaut), indicating that the vowel is considered a separate vowel, even though it would normally be considered part of dipthong. cooperation