Both terms mean the exact same thing. There is no difference between jailbroken and iPhones with stock firmware. "Home screen" is what people usually say. It's general, and easy to understand. "SpringBoard" is the same thing, but it's the technical term for it. "SpringBoard" is also the actual process that is running on the iPhone that is kind of the backbone of the phone. It keeps everything running. Similar to Finder on a Mac or Windows Explorer on a PC.
In short, the two terms mean the EXACT same thing. Some people just prefer to use one term over another.