Yes, they are synonyms. You can use them interchangeably.
Strictly, "horrible" means "inspiring horror," "terrible" means "inspiring terror," and "awful" means "inspiring awe." "Awe" is not NECESSARILY negative. In OLD poetry and writing you may see things like "the awful majesty of Niagara Falls."
There is a classic kids' book by Judith Viorst entitled "Alexander and the Terrible, Horrible, No Good, Very Bad Day."