These two map formats look very similar (based on SQLite and Tiled Maps).
I am looking for method to convert maps from mbtiles to RMaps SQLite and vice versa without quality lost.
A reasonably easy to use GUI based tool for this is Mobile Atlas Creator: http://mobac.sourceforge.net/ I've used it many times to do this very operation.
Just add your existing MBTiles file using "SQLite based atlas formats" instructions at the very bottom of the manual here: http://mobac.sourceforge.net/MOBAC/README.HTM#FileBased
And create a new RMaps format output file. All it does is copy the tiles from one SQLite container to the other, so no loss of quality.