Both are correct. Do not worry : ) 休日 sounds a little more formal to me, or probably to us, but is very commonly used in everyday life. In Japanese Wikipedia, 休日 is used as the title of the holiday page. http://ja.wikipedia.org/wiki/%E4%BC%91%E6%97%A5

I tend to look at these things through my English prism, which often has Germanic/Latinate pairings, the former being the "earthier" word and the latter the more "sophisticated" (car/automobile, dog/canine, sun/solar, etc.). So, too, in Japanese where the Japanese Japanese "休みの日" is "earthier" and the Sino-import "休日" is slightly higher-brow. Not a matter of correctness, but sometimes one of register. As a non-native, I don't feel a big diff here, but the Sinoversion is shorter, obviously, so it might show up more often on spatially challenged signs :)