I am a big fan of cowboy music and western and old timey music. I am less a fan of country/western or country music as generally performed today. My definitions are largely my own as I collect, classify and play recordings.

Cowboy music is, to my mind, the work song...cattle drives, ranching, rodeoing, etc. Tex Ritter, Don Edwards exponents of this.
Western music conveys the beauty and granduer of the western outdoors. Think Sons of the Pioneers and others. A sub grouping of this might be B-western music--altho' lots of that is composed music for the films.
Old Timey, or Hillbilly, music is a rural music, east or west, often using jury-rigged instruments along with conventional ones.
Country/western is pop music with a western feeling, but filled with lush strings and/or lots of brass and reeds. Dean Martin, Jerry Wallace are the worst offenders.

Again, these are parameters I use strictly for myself; there are many really good songs which are difficult for me to pidgeon hole...the great miscellaneous.