[FEN ""] 1.e4 e5 2.Nf3 Nc6 3.Bc4 Bc5 4.0-0
In recent years, the move 4.0-0 has taken over from 4.d3 almost entirely -- in my database of TWIC games, if I look at 2750+ games from the last year, I see 4.0-0 70 times and 4.d3 only 5 times. They mostly seem to transpose into the same positions. This seems to be relatively recent and might be a factor behind the recent popularity of 3.Bc4 (related to the question What happened to the Giuoco Pianissimo? , I turned my comments there into this question).
What's the advantage of 4.0-0 over 4.d3?
In case it helps, I seem to recall someone asking this elsewhere (maybe on Reddit?) and a helpful answer linked to some well known commentators explaining it (perhaps Svidler and Gustafsson?). I was at work and didn't watch the video, and I've forgotten where I saw it.