Nano, if *I* make the decision to do something later than planned, I have delayed. I decided to delay my trip for a few days when I caught a cold. I delayed my trip for a few days.
If something else prevents me from leaving when planned, I have been delayed. I wanted to leave work right on time today, but I was delayed by my boss, who had a question. My departure was delayed by about half an hour.
Does that help?