They refer to locations. Derude is "in the outside world", and derinde is "in the inside area", basically.
Then, would it make any difference if I used "ude" or "udenfor" instead of "derude"?
Ude and udenfor would be synonymous here, just meaning "outside (the house, etc.)" Derude has the additional restriction that the speaker is inside the house and looking out. It's literally "der-ude", "out there" (not "out here", which would be herude).
- Hvad gør du derude? - What are you doing out there?
- Hvad gør du herude? - What are you doing out here?
- Hvad gør du ude/udenfor? - What are you doing outside? (Independent of where the speaker is located.)
Location is "ude", direction would be "ud". There are some adverbs that add an e for location, and no e for direction (for example: hjem/hjemme, ind/inde, over/ovre, hen/henne among others I can't list from the top of my head) which can also take a "her" or "der" to the front (such as "derude" = out there)
How would you translate the sentence "Why are you running out of there?", as in out from a certain place?
I would word it "Hvorfor løber du ud derfra?", but I'm not a native Danish speaker, so could be wrong.