"En dag" is used more often when you talk about something that is to happen "one/some day in the future". "Et døgn" is pretty much only used when you want to emphasize that you are talking about a time period of 24 hours. Example 1: "Deigen tar et døgn å heve" - "The dough takes 24 hours to rise" Example 2: "Vi burde dra til stranden en dag" - "We should go to the beach some day"
Same in Polish, dzień/day/denj/dag in law meaning is some day like a monday, a post office delieves package on 3 days, does nit mean in what clock's time. Doba/24 hours/doba/døgn means litterally in 24 hours, the count must accept or reject proposal in 24 hours of receipt.
Even after reading all the comments, I'm still a little confused. If "et døgn" is "24 hours", then is "en dag" something else? Is "en dag" specifically the period from sunrise to sunset? Is "et døgn" specifically midnight to midnight (a calendar day), or could it be 10:30 to 10:30 the following day? Also, is "et døgn" a period of 24 consecutive hours, or just any 24 hours? Sorry in advance for so many questions.