Our hospital nursery uses swings for the newborns. As long as you have the swing reclined and the baby's head supported well (use rolled up receiving blankets), you can use it right away. Some babies don't like it right away, though, so if it doesn't go over well the first time, try again in another week or two!
I don't know about the BRU vs. Target swings, but don't get too hung up on features. The baby doesn't care. Get something with a recline, at least 2 speeds, some sort of music, and batteries (vs. wind up). At least that was the essential list to me. You might want to look at the Fisher Price cradle swings - they are a little more suited for infants than the Gracos, because of the position of the seat.