The main difference between
New Document command in Ribbon and so-called
+New control is that
+New control is connected to Office Web Apps, opens
Create a new file dialog box on click, and gets replaced by
+New document as soon as content type management is enabled in the library.
There's a KB article on this behavior.
If you want to change that behavior, follow @PaulLucas's advise or consult this article at support.office.com - simply turn on content types.
I prefer to think of
+New control as of something more suitable for simple use cases - no defined content types, and nothing like IRM, information policies, customized templates. Just a friendly way for users to get used to Word/Excel Online.