As @Vadim already mentioned in his answer, it's all about the rendering mode of the selected view. However, Vadim is slightly off about the JSLink part, so let me clarify things a bit.
SharePoint 2013 introduced a new feature called "client-side rendering" (CSR), while SharePoint 2007/2010 only offered "server-side rendering" (SSR).
By default every newly created view on a list is using CSR in SP2013. However, there are several ways to switch to SSR. One option, as Vadim mentioned, is to change the WebPart properties or use server-side code to force a WebPart to use SSR.
Another option is to customize a view with SharePoint Designer (by converting to an "XSLT DataView").
A third option, and obviously the less known one, is to create a new standard view via a Web browser and switch the view style to anything other than "Default".
This will force the view to become SSR, because all those view styles are actually XSLT-based (as they have been in SharePoint 2007/2010 already).
The indicator if it is a SSR or CSR is, as you already discovered on your own, is how and where the "Add document" / "new document" option is displayed.
"Add document" at the bottom of a WebPart view is indicating a SSR, while "new document or drag files here" is an indicator for a CSR.
The part where @Vadim is slightly off is that even if you are customising a CSR with JSLink it will not turn your view into a SSR.
Your first WebPart has either been switched on the properties pane to become server rendered, or you applied a customized view.