What is the difference between ‘combining characters’ and ‘grapheme extenders’ in Unicode?

They seem to do the same thing, as far as I can tell – although the set of grapheme extenders is larger than the set of combining characters. I’m clearly missing something here. Why the distinction?

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The Unicode Standard, Chapter 3, D52

• Combining character: A character with the General Category of Combining Mark (M).
• Combining characters consist of all characters with the General Category values of Spacing Combining Mark (Mc), Nonspacing Mark (Mn), and Enclosing Mark (Me).
• All characters with non-zero canonical combining class are combining characters, but the reverse is not the case: there are combining characters with a zero canonical combining class.
• The interpretation of private-use characters (Co) as combining characters or not is determined by the implementation.
• These characters are not normally used in isolation unless they are being described. They include such characters as accents, diacritics, Hebrew points, Arabic vowel signs, and Indic matras.
• The graphic positioning of a combining character depends on the last preceding base character, unless they are separated by a character that is neither a combining character nor either zero width joiner or zero width non- joiner. The combining character is said to apply to that base character.
• There may be no such base character, such as when a combining character is at the start of text or follows a control or format character—for example, a carriage return, tab, or right-left mark. In such cases, the combining characters are called isolated combining characters.
• With isolated combining characters or when a process is unable to perform graphical combination, a process may present a combining character without graphical combination; that is, it may present it as if it were a base character.
• The representative images of combining characters are depicted with a dotted circle in the code charts. When presented in graphical combination with a preceding base character, that base character is intended to appear in the position occupied by the dotted circle.

The Unicode Standard, Chapter 3, D59

• Grapheme extender: A character with the property Grapheme_Extend.
• Grapheme extender characters consist of all nonspacing marks, zero width joiner, zero width non-joiner, U+FF9E, U+FF9F, and a small number of spacing marks.
• A grapheme extender can be conceived of primarily as the kind of nonspacing graphical mark that is applied above or below another spacing character. zero width joiner and zero width non-joiner are formally defined to be grapheme extenders so that their presence does not break up a sequence of other grapheme extenders.
• The small number of spacing marks that have the property Grapheme_Extend are all the second parts of a two-part combining mark.
• The set of characters with the Grapheme_Extend property and the set of characters with the Grapheme_Base property are disjoint, by definition.

The difference in actual usage is that combining characters are defined as a General Category for rough classification of characters and grapheme extenders are mainly used for UAX #29 text segmentation.

EDIT: Since you offered a bounty, I can elaborate a bit.

Combining characters are characters that can't be use as stand-alone characters but must be combined with another character. They're used to define combining character sequences.

Grapheme extenders were introduced in Unicode 3.2 to be used in Unicode Technical Report #29: Text Boundaries (then in a proposed status, now known as Unicode Standard Annex #29: Unicode Text Segmentation). The main use is to define grapheme clusters. Grapheme clusters are basically user-perceived characters. According to UAX #29:

Grapheme cluster boundaries are important for collation, regular expressions, UI interactions (such as mouse selection, arrow key movement, backspacing), segmentation for vertical text, identification of boundaries for first-letter styling, and counting “character” positions within text.

The main difference is that grapheme extenders don't include most of the spacing marks (the set is actually smaller than the set of combining characters). Most of the spacing marks are vowel signs for Asian scripts. In these scripts, vowels are sometimes written by modifying a consonant character. If this modification takes up horizontal space (spacing mark), it used to be seen as a separate user-perceived character and forms a new (legacy) grapheme cluster. In later versions of UAX #29, this was changed and extended grapheme clusters were introduced where most but not all spacing marks don't break a cluster.

I think they key sentence from the standard is: "A grapheme extender can be conceived of primarily as the kind of nonspacing graphical mark that is applied above or below another spacing character." Combining characters, on the other hand, also include spacing marks that are applied to the left or right. There are a few exceptions, though (see property Other_Grapheme_Extend).

Example

U+0995 BENGALI LETTER KA:

ক

U+09C0 BENGALI VOWEL SIGN II (combining character, but no grapheme extender):

ী

Combination of the two:

কী

This is a single combining character sequence consisting of two legacy grapheme clusters. The vowel sign can't be used by itself but it still counts as a legacy grapheme cluster. A text editor, for example, could allow to place the cursor between the two characters.

There are over 300 combining characters like this which do not extend graphemes, and four characters which are not combining but do extend graphemes.