According to ODO, mini- is classified as a combining form. How exactly is this different from a prefix (or an affix, in general)? Can combining forms also be prefixes?
Generally affix is used for morphological pre- and suffixes. So the /-z/ in Who goes there? is an inflectional affix (a suffix, in fact); this is also true of derivational affixes like -ation, -ize, -ism, be-, en- (and -en).
A combining form, on the other hand, refers prototypically of an allomorph of a (usually free) morpheme (like Malay satu /satu/ 'one') which appears when combined with a different morpheme as a word (like se- /sə/ in sepuluh 'ten'; compare dua 'two' and dua puluh 'twenty').
Mini- is a special case, what's called a libfix — a combining form like net- or -pocalypse that has escaped its morphological bonds and now pursues a new career in neologism creation.
The term combining form was probably meant for learners of English; it is not used on the same level as affix, although it signifies a kind of affix. It is commonly used to indicate nominal and verbal stems of Greek and Latin words used (mainly) as affixes in English. (An affix is meaningful sequence of letters or sounds that can be attached to other sequences of letters/sounds to form a word, but which is not an independent word itself with that particular meaning.)
The sequence of letters -ceive, for example, could be said to be meaningless on its own, because it cannot mean anything in English without some other element, like re- or per-. Then it is not considered a morpheme. (A morpheme is the smallest part of a word that still has an identifiable meaning and can be combined with other morphemes.) According to this approach, combining forms cannot exist as morphemes.
However, if you consider -ceive to mean something like "be the entity that something moves towards", or whatever seems to fit, then you treat it as a morpheme, and you say that receive consists of two meaningful parts, however vague or changeable their meanings may be. Because -ceive in receive exists as part of a word, not as an independent word, it is then a suffix there. (A sequence of letters/sounds can be an independent word in one sentence, and an affix in another; but -ceive is never an independent word, obviously. Or, at least, not yet.)
If you take the latter approach, you can use the word combining form for -ceive, as a special kind of suffix. However, the question remains, how are they special? The answer is: they are not really special linguistically. But for learners, it can be convenient to think of morphemes not in terms of affixes and independent words/stems, but in terms of "the form x is related to other forms, but it is only used in a certain situation, namely as an affix". That is more or less what "combining form" supposedly means, although it is apparently only used for Greek and Latin morphemes.
In the English word photography, for example, both photo- and -graphy are commonly called combining forms by those who use the term. Photo- can occur independently—although with a different meaning, so it is arguably a different morpheme there—, but -graphy normally cannot; it can occur with other prefixes, though, such as epigraphy and calligraphy. Then there are also photographic, telegraph, and telegram, which you could say contain allomorphs of -graphy, namely -graphic, -graph, and -gram. (An allomorph is an alternative form of the same morpheme, so with the same meaning and a similar, though different, form.) However, perhaps -graph- should be considered a morpheme, and -y a suffix signifying an occupation or system, -ic signifying an adjective. Then -gram- and -graph- can be considered allomorphs, as they were certainly in Greek, where the difference was merely related to pronunciation, not meaning—in so far as the two can be separated.
As you see, combining form leads to all kinds of problems, and it is a bit of an amateurish term. If you choose to treat certain parts of words as morphemes, it can be practical to present them as belonging to a fixed, clear-cut, special class; but this class will not sustain deeper scrutiny.
[Edit:] As to allomorphism, this occurs in affixes and non-affixes alike (see examples below). Of course the fundamental problem is: when are two sequences of letters to be considered allomorphs, and when entirely separate morphemes? Usually we take etymology into account for allomorphs, as in naming -graph- and -gram- allomorphs in English, or -th and -s in doth/does, or -ion/-ation, or flam- and flamm- in inflame/inflammation, or after- and aft- in afterthought and aft-deck, or in- and im- and il- and ir- in inhumane/impractical/illicit/irrespective, etc. Ultimately, it is a choice, not an immutable fact.
There's a couple of distinguishing factors. A combining form is a modified version of another word (e.g. mini- for miniature) which can be joined to other words (mini + dress), combining forms (photo + graphy), or affixes (cephal + ic).
So a combining form has another form that it derives its meaning from (e.g. electro- and electric), whereas an affix is a modifier without a corresponding base word (e.g. the un- in unhappy is just a negating prefix). This also means that an affix can be attached to a word or combining form, but not another affix.
Combining forms are also often borrowed combining forms from Greek or Latin (-phile, -ography).
For example, "post-" would be a prefix, since it doesn't have a form which stands on its own, and needs to be attached to a full word for its meaning (after, behind) to come into effect.
(Edited to incorporate additional discussion from the comments.)