"Dreadful" means the same thing as "horrible" and "terrible," but it has a slightly different feel.
To me, "dreadful" sounds a little more polished than the other two. Because it is less commonly used, it might sound a little more emphatic, too.
For example, if you said, "I've just woken up and I look dreadful!" someone might think, "Wow, she must look really bad!" ^^
So I would use "dreadful" when you want to mean something is extra horrible/terrible, but you want to say it in a nice-sounding way.