They are quite close in meaning. "Day by day" refers to process and the passage of time, such as the famous dictum of Emile Coué: "Day by day, in every way, I get better and better". Other examples:...
If it's thunder, I try to get my dog to come out from under the table. Or you could go reenact that part in Singing in the Rain. or you could read, draw, write, get off your computer, paint, clean...