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I’m not sure where you’re downloading from, so can’t answer this.
The TDB file is used by MapSource. If you want to install a map into MapSource, you must have a TDB file.
The MDR file contains a searchable index so that you can search for POIs. See more here: http://wiki.openstreetmap.org/wiki/OSM_Map_On_Garmin/IMG_File_Format
I think you’re confusing a few things here.
OSM is just a geospatial database. What you see on the OSM website is a rendering of that data, by default done using a piece of software called Mapnik. Mapnik is not OSM, it is just a renderer. If you click the + sign in the top right, you can change renderer (e.g. Osmarender). Each renderer produces a different looking map because it decides what to show, what not to show, and how to show it. You could go to the Cloudmade website and see yet another version of the same underlying OSM data (and indeed, on Cloudmade you can really get your hands dirty and change the style to your heart’s content).
mkgmap takes OSM data, and creates IMG files. You can then load these into MapSource or onto your GPS. mkgmap uses style files to decide what to show and what not to show. You can edit these style files to suit your personal needs, or indeed to emulate Mapnik, if you wanted to. But that’s no small task!