What is the difference between the suffixes -ize and -ify?

Courtesy of @DanBron, here's what it says in Word Formation in English,
with interpolated translations:

-ify
This suffix attaches to three kinds of base word:
1. monosyllabic words
2. words stressed on the final syllable
3. words stressed on the next-to-last syllable that end in unstressed /i/

Neologisms usually do not show stress shift, but some older forms do
(húmid ~ humídify, sólid, solídify).

Translation: there are restrictions on what kind of words -ify can go on. They have to sound right. Examples of each kind of word root with -ify:

1. pacify, crucify, gentrify (two Latin words and a neologism, all from monosyllabic roots)
2. solemnify, malignify, machinify (likewise 2 Latin, one neo; but these are rarer)
3. shabbify, rubify, calcify (ditto)

These restrictions have the effect that -ify is in (almost) complementary distribution with the suffix -ize.

Translation: the set of roots that can take -ify and the set of roots that can take -ize are (almost) disjoint sets. There is one intersection, which will be described next.

The only, but systematic, exception to the complementarity of -ize/-ify can be observed with (2), the base words in /i/, which take -ify with loss of the final /i/ (as in Nazify /'natsəfai/), or take -ize with no accompanying segmental changes apart from optional glide insertion (as in Nazi-ize /'natsi(y)aiz/).

Translation: you can do it either way with words stressed on the next-to-last syllable that end in unstressed /i/. But they're the only ones.

Semantically, -ify shows the same range of meanings as -ize and the two suffixes could therefore be considered phonologically conditioned allomorphs.

Translation: -ify and -ize can be considered "the same morpheme", just like a and an.

-ize
Both -ize and -ify are polysemous suffixes, which express a whole range of related concepts such as locative, provisional, causative/factitive, resultative, inchoative, performative, similative.

Translation: there are a lot of kinds of meaning (poly-semous < Gk 'many meanings') for the suffix. These types have technical names. Examples follow (I've changed a few old-fashioned terms to protect the innocent).

Locatives are paraphrased as 'put into X, as in computerize, hospitalize, tubify.
- Patinaize, fluoridize, youthify are provisional ('provide with X'),
- randomize, functionalize, humidify are causative ('make (more) X'),
- carbonize, itemize, trustify, and nazify are resultative ('make into X'),
- aerosolize and mucify are inchoative ('become X'),
- anthropologize and speechify are performative ('perform X')
- cannibalize and vampirize can be analyzed as similitive ('act like X').

The suffix -ize attaches primarily to roots ending in an unstressed syllable and the resulting derivatives show rather complex patterns of base allormorphy.

Translation: A lot of funny things happen to the roots when adding -ize. Examples follow.

For example, bases are systematically truncated (i.e, they lose the rime of the final syllable) if they are vowel-final and end in two unstressed syllables (truncated memory ~ memorize vs untruncated consonant-final hospital ~ hospitalize). Furthermore, polysyllabic derivatives in -ize are not allowed to have identical onsets in the last two syllables, if these are unstressed. In the pertinent cases truncation is used as a repair strategy, as in feminine ~ feminize and emphasis ~ emphasize.

Translation: most of the non-neologisms were formed in Latin, under Latin phonological and morphological rules, which don't apply to English. So lots of the roots are not really English words, but rather are related to English words, and it gets very complicated trying to account for all the differences between the roots and the words.

This is one example of how complex even a teensy-weensy part of English grammar can get.
And this isn't even syntax.