Although not an exact translation, cp. En. "to write up" (summarize, notice; list).
The connotatiin in auf "on" completely subsumes that and leaves nothing but the denotional sense, "onto [paper]; [to write] down", at first sight, although the usage is still very much within the realm of drafting a record.
Mitschreiben is used chiefly in the denotional sense, too, i.e. "along, in synchrony"; cp. eg. mit fahren, mitkommen. The etymologic connection to Greek meta might imply something more, and Lat. mit- "to send" (cf. message) might be informative on top of that, however long forgotten if that were the case, not evident from usage at all. Compare nonetheless Mitteilung to that effect; not here: vermitteln (cf. middle, mediate); probably not here: mitgeben, cp. Mitgift "dowry" opposite of "brideprice" (again there's a Greek comparison, viz. "to pay", iirc). Since auf can mean "on top of", a sense like "add to [a listing, chrinical, etc.]" also comes to mind, e.g. in police lingo: to write somebody a notice for a misdemeanor, but colloquially paraphrased when a particular listing is concerned, e.g. dazuschreiben.
At that it's notable that writing used to be a special craft with idiosyncratic terminology.
However it came to be today the main difference is that Mitschrift requires a lecture or other non-written source to copy from.
Whereas aufschreiben may be creative, perhaps spontanious, and so generally applicable that it is not specialized enough to have a corresponding result noun. There's only a superficial comparison in Aufschrift(cp. Überschrift, Untertitel, Inschrift). One might contrast Abschrift, that has very narrow usage in law denoting legal copy of a writ, next to ausfertigung with a similar sense but different legal requirements (ex perfection?).. Another connotation of aufschreiben (of debts, guilts) can be contrasted with abschreiben "to write off" (cp. streichen "to strike, erase from a list", refl. sich ein-/ausschreiben in university "im-/exmatrikulate"). Whereas the connotation of originality can be contrasted with another sense of abschreiben, that is "to write off of, cheat by copying, immitate".
Compare by the way: English copy-editing (a genre of advertising text, very descriptive)? Auftrag "task, mission" also " coating, layer on-top"? Annonce "(ad) copy" versus announce? Apostal, apocryphe, apo-, post (lol what?), En. off, irregular Ger. uff "up, on" from *op, cp. hör auf "stop" (listen up; leave it), contrast weg "away, off", note auf-g'- (*i.e. pastpart. aufgeschrieben)? fax, from facsimile, fac- "do" + simil(ar), suppletive present tense of fio, future "to be(come)", cf. (last) will? cp. of-ficial???