"Exact differences" only exist between languages, and you absolutely cannot rely on a person's writing system as a resolution of how things are pronounced. First, most linguistic material is not created by highly-trained ear phoneticians with classical Edinburgh-type knowledge of IPA letters. Second, even looking at material created by such scholars, the IPA has only a certain granularity, so you cannot write "exact differences" between languages. You can perhaps get "close enough".
Any of [mʱ m̥ mʰ] are good enough for denoting "that phoneme", and all three are used in some work. For example, in Klamath [m̥] was used because it was easy (in the typewriter days); [mʱ] tends to be popular with people having certain ideas about phonetics; Gray uses [mʰ] for the Tanzanian language Kisi. Given your interest, the most important thing to realize is that written representations of languages are generally not reliable reproductions of how things sound, unless there is an accompanying detailed description of pronunciation. I might therefore describe for you how "mh" in Gogo sounds different from "mh" in Shona or Lhasa Tibetan, though I would prefer to refer you to sample recordings (sorry, I'm fresh out).
Given that IPA is inadequate for conveying the facts of pronunciation about languages, unless you invent a new system with 10 times the number of letters (and can provide reference values for all of those letters), you will still face the granularity problem, that any writing system can only get so close to actual pronunciation.
For your three choices, the first has a breathy voiced release; the second lacks vocal fold vibrations during its production; the third has "aspiration". If you find two differences being contrasted in Ladefoged and Maddieson's book, you can take that as an indication that there could be is a meaningful claim underlying a choice of spelling. The only language that I have encountered with actual voiceless nasals is Angas, where in the production of utterance-final nasals (and liquids), the vocal folds stop vibrating in the middle of the consonant (that's more a "have to see it" than "can hear it" feature). They do not sound at all like "m̥" in Tibetan; Tibetan "m̥" (romanized as "mh") also does not sound like Shona mh which is breathy throughout (thus [m̤], which you didn't ask about).