The difference between these types of wallpaper is the method of applying or activating the paste before hanging the paper.
When using Paste-the-Wall wallpaper, the paste is applied directly to the wall before hanging each strip. This wallpaper is made with a special non-woven backing paper that does not expand when wet; therefore, it does not need to soak as in other application methods.
When using the Paste-the-Paper method, the paste is applied to the back of the wallpaper before hanging. This method requires soak time, which is a way to activate the paste while letting the paper rest before hanging the strip.