Why do we use democracy vs. demarchy and anarchy vs. anocracy?
According to Liddell & Scott, An Intermediate Greek–English Lexicon (1889), the root word κρατος (kratos) in ancient Greek meant "strength, might"—and more generally, "power" or "rule, sway, sovereignty." The same lexicon reports that αρχη (arche) meant "a beginning, origin, first cause," but also "the first place or power, sovereignty, dominion, command." The Greek roots thus have considerable overlap, but the connotations of αρχη may have included a stronger sense of hereditary or historical primacy than those of κρατος.
Today both -archy and -cracy are centrally associated with the idea of ruling. Michael Quinion, Ologies and Isms: Word Beginnings and Endings (Oxford, 2002) has this to say about the suffixes:
-cracy Also -crat, -cratic, and -cratical. Government, rule, or influence [Greek kratia, power or rule.]
Many forms ending in -cracy have been coined, though only a small number are at all well known; most can mean either a system of influence or rule or a society o ruled, as with democracy, rule through elected representatives; a few can also refer to the rulers as a group, as with aristocracy (Greek aristos, best), rule by members of the highest social class.
-archy Also -arch. Government; rule of a particular type; a chief or ruler. [Greek arkhes, ruler; arkhein, to rule.]
Words in -archy are abstract nouns for types of government, leadership, or social influence or organization. They correspond to nouns in -arch for a person or people who rule or command in that way. For example, a monarch (Greek monos, alone or single) is a sovereign head of state, in a type of government called monarchy.
Among the words Quinion lists in a table headed "-cracy Government, rule, or influence," are these:
aristocracy, autocracy, bureaucracy, democracy, gerontocracy, meritocracy, mobocracy, plutocracy, theocracy
Among the words that Quinion lists in a table headed " -archy Government or rule" are these:
anarchy, autarchy, hierarchy, matriarchy, monarchy, oligarchy, patriarchy, synarchy
Because many English speakers when presented with the suffix -archy may think first of monarchy and when presented with the suffix -cracy may think first of democracy, they may suppose that a stronger distinction exists between the tendencies of the two suffixes than actually exists.
The range of types of rule or government comprehended by English words with each ending is extremely broad, and I don't see any pattern in the choice of one or the other that offers a reliable basis for predicting which ending a new form is more likely to adopt. For example, "rule by children" might on the model of gerontocracy be styled paidocracy, or on the strength of matriarchy and patriarchy be named paidarchy.
Also, Quinion's definitions find very little difference between autarchy ("a system of government by one person with absolute power") and autocracy ("rule by one person with absolute power").
The suffixes have different meanings:
-archy means rule: Monarchy: rule of one, anarchy: no rule, oligarchy: rule of few
-cracy means government: Democracy: government of the people, meritocracy: government by merit
You could say "demarchy", which would mean "rule of the people", or "rule by the people", but you wouldn't be saying what you'd want to say. A democracy is not ruled by people, but rather elected representatives. Or, in a pure democracy, there is no ruler at all, but rather everything is voted on by everyone.
Likewise, an anocracy could be a totalitarian dictatorship (it says nothing of rulers; just that there is no government). For example, Somalia could be called an anocracy because they have no government. But that doesn't mean that people are free to do what they please; they still face the wrath and control of local warlords and other rulers.
The only case I have seen where both endings are in use is autocracy vs autarchy.
The roots κράτος might, force, and ἄρχω, to lead, be first, begin, suggest different emphases:
An emphasis for -cracy on where the power lies and how the power is maintained, and on the use of force to establish a system of government (war/revolution);
An emphasis for -archy on leadership and position, as in a word like hierarchy and cognates like arch.
Intuitively, as a native speaker who has learned primarily from usage rather than reading dictionaries and etymologies, I feel that -archy represents something more formal and structural in nature, while -cracy is more descriptive and less rigid it its application - viz. -cracy is about the flow of power but -archy about the description of positional relationships.
These intuitions seem to apply generally but be recognizable specifically in how one views the contrast pair with auto- (self, by one's self).