What is the difference between a dieresis and an umlaut?

Bryan Garner, Garner's Modern American Usage, second edition (2003) offers the following succinct discussion of the two punctuation marks (which are really one punctuation mark with two different names and functions) that the posted question asks about:

umlaut; diaeresis. These words denote the same mark consisting of two raised dots (¨) placed over a vowel, but they serve different phonetic functions. An umlaut (pronounced /oom-lowt/) indicates that the vowel has a modified sound especially in German, as in Männer (pronounced /men-ner/) A diaeresis (pronounced /dI-air-ə-sis/ and sometimes spelled dieresis) indicates that the second of two adjacent vowels is pronounced separately, as in naïve.

But the distinction is largely academic: even with modern word-processing capabilities, these marks are often omitted.

Allan Siegal & William Connolly, The New York Times Manual of Style and Usage, revised edition (1999) notes that the mark occasionally appears in imported proper names in which it serves a third purpose: to signal "pronunciation of a normally silent final consonant":

accent marks are used for French, Italian, Spanish, Portuguese and German words and names. ...

...

The umlaut modifies vowel sounds in German, (Götterdämmerung, Düsseldorf). Some news wires replace the umlaut with an e after the affected vowel. Normally undo that spelling, but check before altering a personal name; some individual Germans use the e form. In the Latin languages, the umlaut is known as a dieresis. It denotes separated pronunciation of two adjacent vowels (naïve, Citroën, Noël), or signals pronunciation of a normally silent final consonant (Saint-Saëns, Perrier-Jouët).

According to the Wikipedia article for dieresis, the dieresis is used to indicate that two vowels are to be pronounced separately, rather than combined as a single combined sound. On the other hand, an umlaut is used to indicate a sound shift of the vowel, pronouncing it differently than it would without the umlaut.

The diaeresis and the umlaut are diacritics marking two distinct phonological phenomena. The diaeresis represents the phenomenon also known as diaeresis or hiatus in which a vowel letter is pronounced separately from an adjacent vowel and not as part of a digraph or diphthong. The umlaut, in contrast, indicates a sound shift. These two diacritics originated separately; the diaeresis is considerably older.

The dieresis is native to English, though it is generally considered to be archaic. On the other hand, the umlaut is a diacritic found in other languages (in particular, Germanic ones).

The grave accent and the diaeresis are the only diacritics native to Modern English (apart from diacritics used in loanwords, such as the acute accent, the cedilla, or the tilde). The use of both, however, is considered to be largely archaic.

Germanic umlaut is a specific historical phenomenon of vowel-fronting in German and other Germanic languages. In German it causes back vowels /a/, /o/ and /u/ to shift forward in the mouth to /ɛ/, /ø/ and /y/, respectively. In modern German orthography, the affected graphemes ⟨a⟩, ⟨o⟩ and ⟨u⟩ are written as ⟨ä⟩, ⟨ö⟩ and ⟨ü⟩, i.e. they are written with the diacritical mark "umlaut", which looks identical to the diaeresis mark.

Some languages have borrowed some of the forms of the German letters Ä, Ö, or Ü, including Azerbaijani, Estonian, Finnish, Hungarian, Karelian, some of the Sami languages, Slovak, Swedish, and Turkish. This indicates sounds similar to the corresponding umlauted letters in German.

[Diaeresis - Wikipedia]

The simplest summary is: Umlaut is a German phenomenon, while dieresis is used mostly in French (see examples above).

At a closer look they look similar but convey totally different meanings: Dieresis indicates that two vowels are to be pronounced separately, whereas umlaute originally symbolized ae (ä), oe (ö), or ue (ü). And these vowel combinations actually are vowels of their own in the German language (see above for pronunciation guidelines).

Cheers, Günther

P.S. Actually the dieresis is called a "Trema" in German and most Germans don't know about it before starting to learn French.

Other answers have addressed the question pretty well in a general context, so I thought I'd give an answer focussing more on the linguistic history in particular

In this context, the diacritic is not referred to as an umlaut, but rather a diaresis (if indicating that the letter should be pronounced separately, rather than as part of a digraph) or a trema, whilst umlaut refers to a set of partially-grammaticalised partially-lexical vowel changes in the Germanic languages, some of which are shown with a trema in German orthography

There are three types of umlaut, i-umlaut, a-umlaut, and u-umlaut. The first two occurred in all surviving Germanic languages, whilst the latter only occurred in the North Germanic (Scandinavian) languages

I-umlaut affected vowels followed by an i (or a j, which in most Germanic languages has the sound of an English y). Back vowels (a, o, & u) affected by it move forward in the mouth (creating new vowels ä, ö, ü, in English these later became e, e, & i), whilst front vowels (e & i) move up in the mouth if possible (merging into i)

A-umlaut affected vowels followed by an a, and only affects the high vowels (i & u) which were lowered to merge into the mid vowels (e & o)

U-umlaut affected vowels followed by a u (or a w, which later became a v in most Germanic languages). Unrounded vowels (i, e, a) affected by u-umlaut became rounded (ü, ö, å)

In all cases, long vowels behaved pretty much the same as their short equivalents

If this was just easily predictable though, it wouldn't have ever needed to be written explicitly, you'd just know to pronounce an a followed by an i further forwards in your mouth. The reason it came to be written explicitly is because these vowel changes persisted after the vowels that triggered them were lost

As an example, the English word man has the plural men, reflecting an earlier stage where it was mann and manniz with the i triggering i-umlaut. The -iz ending was lost, but the umlaut from a > ä (which later developed into e) was retained

German reduced unstressed i to e in most cases, so most instances of i-umlaut that still had an obvious trigger seemed to have it triggered by an e, and so German scribes started writing an e above a vowel to show that it had been affected by i-umlaut and this eventually evolved into the modern German trema we see today

The diaresis as used in French (to show that the second vowel letter in a sequence should be read separately, rather than as a digraph) is a direct continuation (via Latin) of the Greek trema used in the same way developed during the Hellenistic era (along with most of the rest of the Greek diacritics)

I suggest looking at Babbel.com A Tale Of Two Dots — The History Of The Umlaut And The Diaeresis

The meaning of the word umlaut is revealing: it means “around sound” in German. It was named by the linguist Jacob Grimm, one of the Grimm Brothers. His “around sound” describes a process of sound-change where a vowel’s sound is influenced by another vowel that follows it in the word. The plural of Hant (“hand”) in High German was Hanti, but the i ending influenced the pronunciation of the previous vowel a. So, Hanti became Henti. Eventually, the final i lost its timbre. So, in contemporary German we have Hand and the plural Hände.

The representation of the umlaut in Middle High German was sometimes denoted by adding an e to the affected vowel, either after in regular size or above in a smaller size. The former is still applicable in a name like Goethe — which is never spelled Göthe.

In French, the diaeresis (from the Greek diaíresis [διαίρεσις], meaning “division” or “distinction”) looks exactly the same as the umlaut but has a radically different purpose. Whereas the umlaut represents a sound shift, the diaeresis indicates a specific vowel letter that is not pronounced as part of a digraph or diphthong. In French words such as Noël (Christmas), the two dots are there to remind you not to fuse the two vowels into one sound, but to pronounce the O and the E separately.

In Spanish, … vergüenza (“shame”) or ambigüedad (“ambiguity”), which is usually silent in words such as guitarra (“guitar”) or guionista (“scriptwriter”).

English also has its occasional diaeresis, mostly used in surnames or given names, such as Brontë or Zoë. In some words, such as naïve, it is optional.

The whole article is worth reading.