There should be no difference in meaning or acceptability, any more than there is a difference between "oeconomics" and "economics", or "oestrogen" and "estrogen". As with (o)estrogen, the "oe" spelling is generally preferred in British English, and the "e" spelling is associated with American English. As far as I can tell, the sources you found prefer "homoeopathy" basically as a matter of branding and crankism (not really surprising that this kind of thing would be associated with a pseudoscientific worldview).
There is no actual connection between Latin "homo" and Greek "hom(o)eo-", by the way. The actual Greek root meaning "same" is just "homo-" (ὁμός) (as in homology, homotopy, homosexual), which is also not connected to the Latin word "homo" meaning "man"; both "homoeo-" and "homeo-" are variant spellings representing the same Greek root (ὅμοιος) meaning "similar". So the first source you link to is very confused from a linguistic perspective.
The Oxford English Dictionary lists a third spelling variant "homoio-", but I can't say I've ever seen the spelling "homoiopathy".
It also has a note on the pronunciation:
The etymological pronunciation would be /hɒˈmiːəʊ/, as in /hɒˈmɔɪəʊ/; but usage favours /ˈhɒmɪəʊ/, or in popular use /ˈhəʊmɪəʊ/; the last esp. in homoeopathy and its family (the only really popular members of the group).
While the spelling "homoeo" is more likely to suggest the pronunciation /hɒˈmiːəʊ/, this isn't really useful for "homoeopathy" because it isn't usually pronounced that way (a comparable case is ornithological, which according to the OED's view of these things "should" be pronounced as or-NIGH-thological based on the etymology, but isn't; similarly arch(a)eological, pal(a)eographic are not pronounced as ar-KEE-ological, pa-LEE-ographic despite having long/diphthongal vowels in the original Greek).
History of general variation in spelling between "ae/oe/e"
The Latin language at one point had dipthongs something like [ai̯] and [oi̯] (more or less the sounds of English "eye" and "boy"). In Classical Latin, the corresponding sounds were written "ae" and "oe"; some people think this indicates a pronunciation change that had occurred by this point whereby the off-glide was somewhat lowered, making them something like [aɪ̯] and [oɪ̯] or [ae̯] and [oe̯]. Eventually, they developed in pronunciation to monopthongs [ɛː] and [eː] respectively in Vulgar Latin. (Actually, even from early on there was a tendency, seen as rustic, for Latin "ae" to become a monophthong that merged with "long e", so sometimes "ae" corresponds to [eː] instead of [ɛː] in Vulgar Latin.) I have read somewhere about a hypothesis that "oe" monophthongized before losing rounding, passing through a stage like [øː], but I don't know if there is any particularly good evidence for this, although it seems plausible to me.
Anyway, the [eː] that developed from "oe" was pronounced the same as the [eː] that developed from long "e" or "ae", and the [ɛ] that developed from "ae" was pronounced the same as the [ɛ] that developed from "short e", so by Middle Latin the letter "e" could represent either of the sounds associated with "ae/æ" and "oe/œ". This led to a lot of variation between spellings with "æ/ae" and "e", or "œ/oe" and "e", or in some cases between "æ/ae" and "œ/oe" (e.g. the Classical Latin word "caelum" could be spelled in the Middle Ages as "cœlum").
This variation continued into a number of languages that borrowed words from Latin; in English, the most relevant thing affecting this later on was possibly Webster's preference for spellings with "e" which seems to have contributed to present-day differences in custom between American English and British English.
It has no special etymological significance.
Greek αι was conventionally Latinized as "ae"; Greek οι was conventionally Latinized as "oe", so the same variation usually applies to words we get from Greek. Occasionally, people try to use a less Latin-based Romanization of Greek, and use "ai" and "oi" instead (e.g "Koine", "ailurophobia"). This is not significant either (a variant form "aelurophobia" exists, for example).