Google is very helpful for usage questions, and in this case it suggests that there are differences:
Google hits by form (in millions, searched in kana & kanji)
| -u form | -eru form | ratio (:1) | |
| dictionary | 4.96 | 11.50 | 0.43 |
| nom. | 176.00 | 87.20 | 2.02 |
| pol. (-masu) | 0.11 | 0.52 | 0.21 |
| short past | 20.40 | 23.90 | 0.85 |
| pol. past | 0.64 | 5.11 | 0.13 |
| prov. (-ba) | 0.99 | 1.21 | 0.82 |
| cond. | 0.61 | 1.50 | 0.41 |
| -te | 58.80 | 28.80 | 2.04 |
| short neg. | 1.11 | 4.52 | 0.25 |
| pol. neg. | 0.10 | 0.42 | 0.24 |
| overall | 263.72 | 164.68 | 1.60 |
Only a very hazy index, and of course it's biased toward online use and doesn't tell us anything about regional usage or other forms or specific cases. But it does suggest that some distinctions exist. 間違う forms overall are about 60% more frequent than 間違える forms, and most of the individual differences are pretty marked; in only a few cases are the usage numbers comparable.
The definitions on Denshi Jisho are too general and too similar to help. Goo has much more detail and in fact lists them together as a set of three "quasi-synonyms" (類語, "ruigo"): machigau, machigaeru, and ayamaru, which is usually defined as "to apologize". (Will have to look into ayamaru and its several kanji later.) Here (with ayamari in advance for the many machigai in my rough translation) is what Goo says about them:
GENERAL MEANING (ayamaru / machigau / machigaeru)
to make an incorrect decision or judgment; also, an error or failure in action or behavior [ie, to be wrong about something]
USAGE (examples)
ayamaru: "to misjudge" (as an approx measurement with the eye); "to make a misstep" (physically miss one's footing); "to lose one's way" / "take the wrong path" (ikikata o machigaeru, "err in one's way of living", with machigaeru)
machigau: "the answer isn't wrong"; "that's wrong" (are ha machigatteiru)
machigaeru: "to mistake an umbrella for something else" (kasa o machigaeru); "mistakenly sent an invitation to Mr. Tamura rather than to Mr. Murata" (tamura-shi to murata-shi o machigaete, annaijou o dashita) [so, "to confuse two things" or "to mistake one thing for another"]
USAGE (distinctions)
1. used as a transitive verb, "ayamaru" means to acknowledge an incorrect judgment about something. [ie, to apologize] in the form "...o ayamaru", usually the error or acknowledgment goes in the "o" part. in many cases the form "ayamatta" is used intransitively to modify a noninflectable word (eg, "ayamatta kangae", "mistaken thought"), and used with the sense that one is wrong, said something false, judgment or understanding is incorrect. furthermore, currently there are many cases in which "ayamatte" is used with the sense of "inadvertently"/"accidentally", sort of like an adverb.
2. "machigau" is really an intransitive verb, and there are situations in which the two are not interchangeable. however, currently, it's almost the same as "machigaeru" (eg, "順番を間違う, junban o machigau").
3. as an intransitive verb, "machigau" follows three rules of usage/meaning: (i) when pointing out a person's concrete/tangible "misses" or failures—"man, he messed up again!" (あいつまた間違った!)—the person becomes the subject. in these cases, the focus is on superficial action/behavior. this form is also used to address a lapse in behavior without mentioning the error in judgment. (ii) as with phrases like "your thinking is mistaken / you are mistaken in your thinking" (anata no kangae ha machigatteiru), it's used to mean adverse deeds or behaviors that result from a person's thinking, choices, or judgments; (iii) as with "the answer is wrong" (kotae ga machigatteiru), it's used to mean bad (incorrect) outcomes that result from errors of thought or judgment.
4. the intransitive "machigau" and "ayamaru" are [both] close in meaning to "to misjudge / make an incorrect decision". if you really want to distinguish them (!), "machigau" has the sense of making a more specific/concrete/tangible (gutaiteki) "miss", and "ayamaru" can say something's wrong without specifying the occasion or nature of the error and has a more more abstract feel. so, vs "your thinking is mistaken" (ayamaru), "your thinking is more mistaken" (machigau) has developed the nuance of more direct speech / more of a criticism. also, generally / in popular usage, "ayamaru" is more natural for ideas, cognition, etc., and "machigau" for cases in which concrete outcomes result from mistakes in thinking or judgment.
5. "machigaeru" is really/originally a transitive verb that corresponds to intransitive "machigau". "machigaeru" refers to superficial actions, like missing your fingering while playing the violin or making an error in a calculation or a response to a question—a glitch in thinking that results in an adverse outcome. Also used with the meaning of "to mistake/confuse" (torichigaeru): "confusing an umbrella" (with something else, I guess); "to mistake sugar for salt" (shio to satou o machigaeru).
RELATED WORDS
chigaeru: to form an understanding or judgment or behave in a way that is not normal. also, to (be permitted to) be or behave in a way that differs from other things/people or the norm. "kotae o chigaeru" (miss the answer); "atama no suji o chigaeru" (pull/strain a muscle in one's head); "yarikata o chigaeru" (to do it wrong, to make an error in method).
(That's some tough language for me, so I'll have to take another look at it later vs the original. Caveat lector.)
So the apparent 2-to-1 frequency of machigau in the nominal and -te forms makes sense; probably machigai is more common (as a general "error") than machigae, and machigatte probably appears more frequently in compound use. 塩と砂糖を間違えて料理に置いてしまいました。"Mistaking it for salt, I (disastrously) put sugar on the food."—less specific, maybe, than uses for machigau.
Particles may shed some light, so here's a table of hits for GA+machiga- (subject marker) and O+machiga- (object marker):
| ga | o | ratio | ga | o | ratio | ||
| -u | 0.1 | 1.0 | 0.1 | -eru | 0.2 | 3.4 | 0.1 |
| -tta | 17.8 | 6.4 | 2.8 | -eta | 4.0 | 23.9 | 0.2 |
| -ttara | 0.5 | 0.2 | 2.0 | -etara | 0.3 | 1.5 | 0.2 |
| -imasu | 24.8 | 0.0 | 826.7 | -emasu | 0.1 | 0.5 | 0.1 |
| -imashita | 0.2 | 0.2 | 1.1 | -emashita | 0.6 | 5.2 | 0.1 |
Not as clear as I'd hoped. Overall, machigaeru strongly and consistently prefers o (ratio below 1), across the board, and machigau prefers ga (ratio above 1). But there's a lot of variation in machigau: the pref for ga in the -masu form is waaay higher than for anything else—more than 826-fold—but in the plain (short) form machigau seems to prefer o almost as strongly as machigaeru does. 変ですね。 Even weirder: that intense preference for ga in the polite machigau is not reflected in polite past machigau, which seems to use the two about equally. それを間違いました、それが間違いました。 それを間違えました。それを間違った、それを間違えた。
The Goo page on "machigai" also looks helpful, but that's for another time. And then there's the question of "ayamaru" and "ayamatsu"....and 過 (過つ/過ぎる/過去)....
All this came about originally because I was looking for a way to apologize for errors in an e-mail to 先生 without committing new ones! 間違ったら、どうもすみません。 ？
