The Mendix Business Modeler is just an old name for the Mendix Modeler, so yes they are the same.
There's two kind of modelers:
Starting a new app from SprintR will automatically enable / start the Web Modeler. However, you can always open (and sync) the Web Modeler with the Desktop Modeler.
I'd suggest following the learning paths at http://gettingstarted.mendixcloud.com