What is the difference between the root "hemi", the root "semi", and the root "demi"?

The three prefixes originally had overlap but some different nuances:

"semi-" was used generally and imprecisely: it did mean "half" but with some latitude.

"demi" was used to mean half of a figure/body and also "not quite a [noun]"

hemi- was used for half of a symmetrical object around the line of symmetry. It was more popular in technical terms.

OED:

Semi- prefix:

Etymology: representing Latin sēmi- ... = Sanskrit sāmi- , Greek ἡμι- , Old High German sâmi- , Old Saxon sâm- , cognate with Old English sam-

Demi (adj and noun) Etymology: < French demi < Latin dīmidium half: see dimidiate adj. The French word is a noun and adjective, and much used in combination. It began to be used in English in the 15th cent. attributive in Heraldry, and in the 16th cent. in names of cannon, and soon passed to other uses.

1. In general use.

a. (a) Compounded with adjectives and participles, with the meaning ‘half, partly, partially, to some extent’. A few are used elliptically as nouns.

1871 W. Bagehot in Fortn. Rev. 1 Aug. 158 A semi-abstract discussion of practical topics.

demi-, prefix

Etymology: < demi adj. and n.

Used with the senses ‘half, half-sized, partial(ly), curtailed, inferior’.

1. In Heraldry, etc., indicating the half-length figure of a person or animal, or the half of a charge or bearing. 1486 Bk. St. Albans, Her. B v a Demy is calde in armys halfe a best in the felde.

hemi-, prefix

Etymology: < Greek ἡμι-, combining form, from earlier *σᾱμι- = Latin sēmi- , Sanskrit sāmi- , Old Germanic *sâmi- ,...Several Greek words containing this element were in use as technical terms in later Latin, e.g. hēmicyclium , hēmīna , hēmisphærium , hēmistichium . In the modern languages they are very numerous, not only in terms adopted or adapted < Greek (directly or through Latin), but in new formations, scientific or technical, < Greek, or on Greek analogies. Words formed < Latin have the corresponding prefix semi- prefix; but there are instances of hybridism in the use of both prefixes.

1. Half-; one half, the half, pertaining to or affecting one half; esp. in Anatomy, Biology, and Pathology. Pertaining to one of the two halves (right and left) of the body, or of any of its symmetrical organs.

The infrequent use of these prefixes, their similar applications and sounds, and a general lazy tendency of language to gravitate towards a mean has caused all three to be used either as they were fossilised, or to be used to create words that "sound authentic".

The simple answer would be what gained traction. Some people started using a particular compound word and that word then became widely used and it remained so.

Hemi- is Greek; Semi- is Latin; Demi- is French. They all mean some kind of "half."

This website: mentalfloss.com describes the minute differences in their meanings (hemi,semi,demi).

This answer by @tum_ details the origin of the prefixes in a related question previously asked on SE.

Semi- is Latin, hemi- is Greek, and demi- is French (and ultimately Latin, but from a different root than semi-).

When it doubt, use semi-. It's the most common of the three.

Hemi- is used for geometric (hemisphere, hemispheroid, but not semicircle) and biological (hemichordate, hemimetabolism, hemiparasitic) terms.

Demi- is used for demigod(dess), and obscure military terms (demibastion, demigorge, demilance, demilune).

Multiple halving prefixes are used in British music terminology for 32nd (demisemiquaver) and 64th (hemidemisemiquaver or semidemisemiquaver) notes.

Worth noting that hemisphere has a greek etymology for both the prefix and the main part (ἡμισφαίριον), same for semicircle and latin (semicirculus), that's why you wouldn't mix the prefixes in those cases.