Why are there New maps (odbl) and Old Maps CCBYSA?
Well, Opentstreetmap changed their license in July 2012. As there needed to be quite a lot of contributions to be deleted, the state of the maps was for some time quite troublesome. In most countries and areas the maps under the odbl license are as of 2013 far superior. The old maps are also not having all the newest features (e.g. housenumber address search).
The New Maps Can be found Here - They are suitable for Windows and Linux (Qlandkarte GT)
Maps for Mac OSx (new odbl) can be found Here
The Old Maps - not recommended anymore - (Data from 6. July 2012) Can be found Here
Programs to Display the VeloMap
Besides being shown on (nearly) all Garmin GPS devices - most People will use Garmin Basecamp (v3 or later) or Garmin Mapsource to show the maps on their Desktop and plan tours/trips.
For anyone on the good side of earth Openmtbmap.org maps are also correctly shown in Qlandkarte_GT - QLandkarte_GT is compatible with all big OS (Linux, MacOSx and well Windows too). Linux users can alternatively install Mapsource 6.13.6 in WINE.
For Linux/Unix users there is also a script to unpack the maps, and automatically create a gmapsupp: https://github.com/btittelbach/openmtbmap_openvelomap_linux
Mapsource - I wouldn't use any version but 6.16.3. Mapsource versions 6.14 to 6.15.6 are complete junk, if you use it, please update. Also note that Mapsource versions prior to 6.16.1 do not show all ways and pathes so they are not really usable. Mac OSx users please use Qlandkarte GT or convert maps and use Basecamp 3 or Roadtrip: convert-maps-for-mac-os-x-roadtrip/
There are also maps directly convertet for Mac OSx - scroll down or click on: https://www.velomap.org/download/#velomap-premium-maps-download-in-mac-osx-format-gmap