Origin
The suffix -ic comes from Greek -ikos, while -ical is a combination of -ic and the French suffix -al. Originally, -al was suffixed to scientific nouns ending in -ics, e.g. mathematics - mathematical. Eventually, the -ical portion of those words was reanalyzed as being a single unit. This is what Marchand (1969) had to say about -ic vs. -ical at this time in their history:
There was, at the beginning, indiscriminate coexistence of two synonymous adjectives. But language does not like to have two words for one and the same notion, and competition was bound to come.
What Marchand is talking about is that these two redundant suffixes eventually settled into certain niches where they exist today (more on this later).
-ic and -ical doublets
There exist many -ic/-ical "doublets" in English, where both forms of the word exist and are used regularly.
For example:
- historic and historical
- electric and electrical
The words in common doublets have generally developed distinct meanings (or domains of use). For example, historical refers exclusively to things that happened in the past, while historic can refer to an event occurring at that very moment that will have a profound impact on history.
However, the differences in meanings between -ic and -ical words (in cases where both forms are used) cannot be generalized to a specific unique meaning that -ic or -ical itself contributes to the word.
That is to say, there is no general and consistent meaning that -ical uniquely contributes to historical, electrical, etc. Instead, each word as a whole just settles into its specific meaning.
Distribution Today
According to a systematic productivity measure by Lindsay and Aronoff (2010), the more productive of the two suffixes is clearly -ic, which is favored by approximately an 8 to 1 ratio using this measure. So, you are much more likely to have an -ic form of a word, or to have that form be the more commonly used form.
Now, if -ic is clearly used more than -ical, then why do we have both of these suffixes in the language?
Well, -ical has managed to differentiate itself from -ic, but not by contributing a specific meaning to the words it attaches to. Instead, -ical has found a specific morphological domain in which it attaches very productively: stems that end with the morpheme -(o)log-. Examples:
- biological preferred over biologic
- technological preferred over technologic
- typological preferred over typologic
There are nearly 500 stems in Webster's 2nd dictionary that end in -olog. Using the same measure of productivity, Lindsay & Aronoff found that within the domain of these stems, the -ical suffix was itself preferred by an 8 to 1 ratio over -ic. That is to say, when there is an -(o)log morpheme at the end of the word, then people heavily favor -ical.
The notion of one suffix being productive when attaching to a certain suffix is known as "potentiation", coined by Williams (1981). (Another example of potentiation is what we see with words ending in -able. In general, -ness is considered a more productive suffix than -ity, but in words ending in -able, -ity is strongly preferred.)
Bear in mind that these are tendencies; while there is a strong preference for one form over another in certain cases, that does not mean that it is a perfect split.
Summary
-ic and -ical do not differentiate themselves semantically; however, both forms of a word can exist if each one settles into a different meaning, or otherwise has a different domain of usage (for example, one formal and the other informal). The actual differences in meaning between two forms in a doublet are essentially arbitrary.
While -ic is much more common overall (and could be argued to be the "default"), -ical is strongly preferred in stems ending in -(o)log-. Again, this is a tendency that began somewhat arbitrarily during the development of the language, but because words ending in -olog form such a large and relatively uniform group, a subpattern was able to emerge and stabilize, even as -ical remains in the minority.
The prefixes in- and un- both have the effect of negating the meaning of the word. The prefix in- comes from Latin and almost exclusively applies to words borrowed from Latin, which in many cases were borrowed from Latin with the in- prefix already attached. The prefix un-, on the other hand, is a native English prefix, and so, in general, can apply to any English adjective.
The Online Etymology Dictionary tells us that inequality was borrowed from Latin (via French) with the prefix in- already attached, so we get the word with the prefix in-:
1484, "difference of rank or dignity," from O.Fr. inequalité (14c.), from M.L. inæqualitas, from inæqualis "unequal," from in- "not" + æqualis "equal"
The word unequal, on the other hand, does have the same Latin root of æqualis, but the prefix un- doesn't seem to have been applied to it until after it became an English word.
There exists this phenomenon called blocking, where two seemingly equally valid ways of saying something conflict, so the existence of one blocks the other. Since the word inequality already existed from having been borrowed from French with the negative prefix already attached, the process to form the word unequality was blocked. The word unequality does exist though, but its usage is far eclipsed by inequality.
Best Answer
The etymological origin of the root of the word isn't actually as relevant as whether the ending of the adjective is from Latin or from English.
The negative prefix in- is not very productive in English1: that is, it's rare for a speaker to take an English adjective and attach in- to it to create a new negative adjective. Rather, in- (or im-, ir-, or il-) mainly shows up on words that already existed with this prefix in Latin or French before they were adapted for use in English. For example, before the word instability was used in English, instabilité was used in French.
As John Lawler says in his answer, uncompleted is derived from the past-participle form of the English verb complete, as indicated the by the presence fo the suffix -ed. It was not taken directly from a Latin adjective.
In contrast, complete is the anglicized form of a Latin adjective completus which corresponded to a negative adjective incompletus (listed in Lewis and Short's Latin Dictionary, although as a "late Lat." word).
The reality is a bit more complicated than the preceding paragraphs suggest. For one thing, in- does seem to have had some productivity in English in contexts where it creates a word that looks like it could have come from Latin or French, even if there wasn't actually a preexisting Latin or French word of this form. The word invariant may be an example of this type: the Oxford English Dictionary's earliest citation for the use of the word "invariant" in English is from 1851, which is earlier than the date of 1877 in the Trésor de la langue française informatisé's entry for invariant as a French word.
The OED entry for the negative prefix in- suggests that its usage has changed some over the centuries, and mentions the substantial overlap with the usage of un- which has resulted in some amount of variation for a number of words:
Keep in mind that this entry "has not yet been fully updated (first published 1899)" and it may have some inaccuracies. I think the author may have been overstating things a bit with the wording "freely used": Dixon, who I cited above and link to below, says in- is "seldom used to derive new [words]" (p.73) and categorizes it as "scarcely productive" (p. 73-74).
An important category of exceptions to the rule that I suggested in this section is adjectives that have Latin-derived endings but that nonetheless are negated with un-. Some even have corresponding -ity nouns starting with in-, such as unstable, unable, unequal vs. instability, inability, inequality. Other examples of adjectives with Latin-derived endings but the English negative prefix un- are unusual, unreal, unconscious, unscrupulous, ungenerous, ungracious, unattractive, unexpressive, unresponsive. I don't have stats on the relative frequencies of adjectives like this vs. adjectives like impossible, inaccurate, insensitive, intolerant that have the latinate negative prefix to go with their latinate endings. In general, un- is more common than in- (similar to how -able is more common than -ible), so the ending-based "rule" I suggest might be thought of as a rule for when a word might start with in- rather than a rule for when a word is likely to start with in-. A related question: Why can we use "inadequate" but not "inspecific"?
Words ending in -ed very rarely can be negated by prefixing in-/im-/ir-/il-
According to João Bittencourt de Oliveira, "Past participles ending in -ed take un- [...] Inexperienced is the only past participle with the prefix in-"2. I suspect that the existence of the related noun inexperience is relevant, although I'm not sure exactly how. I think that experienced may not be a past participle, strictly speaking: although there is a verb experience, the suffix -(e)d in English is not only used to form participles or adjectives from verbs, but also to form adjectives from nouns (e.g. "winged" or "armed"), and the meaning of experienced seems more in line with this other usage of -ed. But whatever the internal structure of experienced, the word inexperienced still seems to me to have the structure in- + experienced rather than inexperience + -ed, which is why I'm uncertain about whether the noun inexperience has made any contribution to the use of in- in the adjective inexperienced.
I was able to find one more possible counterexample to the rule that Oliveira mentions: indisposed. The OED also has entries for some other obsolete variant words that are constructed this way, such as indispersed, inaffected, inabstracted. (Of course, a participle derived from a verb starting with in- will also start with in-, e.g. invalidated, but in this case the prefixation occurred before the suffixation.)
So we could say that the use of un- to negate adjectives ending in the suffix -ed is just a tendency rather than an absolute law, but it still seems to be a very strong tendency.
Citation footnotes & links
Making New Words: Morphological Derivation in English, by R. M. W Dixon (2014), p. 73-74.
"Negative Prefixes in Technical and Scientific English", p. 14.