As far as I can tell these are two similar ways to do the same thing. If a launch screen file doesn't offer any advantages over a launch images source then why did Apple offer it? What am I missing?
This answer assumes you're talking about iOS, even though you didn't mention or tag iOS in your original question.
A launch image is simply a static (non-changing) placeholder image.
"Launch Screen" (or more accurately "LaunchScreen" or "LaunchScreen.storyboard"), is the default name of the storyboard for a Launch Screen File; it is a very simple xib or storyboard file that appears before the rest of the app gets loaded up. The advantage to this is that a dev can use auto layout to reposition things subtly depending on the size of the screen (iPhone vs. iPad, iPhone 6 vs iPhone 6+, etc.).