What is the difference between "downloaden", "runterladen" and "herunterladen"?

They three mean exactly the same.

"Herunterladen" (from "Herunter" -> Downwards and "Laden" -> Loading) is the straight german equivalent to "Downloading", although we use the english term with regard to anglicism as verb too when we say "Downloaden".

Both terms are equal frequently used so you can use the one for the other if you like to, although "Downloaden" is a little bit sloppier.

"Runterladen" is the abbreviation of "Herunterladen", more slang and should not be used in official and adequate speaking usually.

What you never should use is the term "Herunterholen" or the abbreviation "Runterholen" (-> Holen = take), which is german slang for male masturbating.

1. "Herunterladen" is the actual German translation for "download".
2. "Runterladen" is just colloquial speech, where "runter" is a lazy reduction of "herunter". Do not use it in official stuff.
3. People often keep the English term "download" as loanword, and just give them the common German verb suffix "en", which turns to "downloaden" then, and it's also just for colloquial speech because it sounds more cool than official. But there is a problem with the integration in German grammar because it consists of "down" and "load". So there are two possibilities, and it's not clear which one will win in the end, both ones are used in similar commonness.
   • In variant 1, we keep "download" together as one word and put the German grammar around that, e.g. "gedownloadet" which wraps the English word with German prefix and suffix (for Partizip II).
   • In variant 2, we look at "download" as two words and split them to get them into German grammar, e.g. Partizip II: "[down]ge[load]et" equivalently to "[herunter]ge[laden]".

There is no difference in meaning.

• "downloaden" is obviously borrowed from English ... however it's a proper loan word, i.e. it takes on forms that are allowed for a verb in German but that would be impossible in the original language
• "runterladen" is the shortened, somewhat sloppy (stylistically speaking), form of "herunterladen"
• "herunterladen" is a literal import of "download" (verb) into German

As far as my experience goes, right now there is no distinction in the german language for that topic.

You have "runter" as a shortened version of "herunter".

Then you have the anglicism "downloaden".

That is all there is nowadays.

So for your case getting it corrected with "runterladen" - you could nitpick him with "herunterladen".