I do not think there is a difference. I believe the ones label ‘Columbia Grafonola’ are just older records. They probably used both names so that people would recognize the name. If you are asking if these were given with gramophones, then yes, they were given away with the machines. The dealers would typically let you pick out an album of records when you bought a machine, but I do not believe it was limited to label or artist.
Columbia introduced the "Grafonola" tag to its logo on record labels in 1917, replacing the older slogan "Note the Notes". You'll find the word "Grafonola" only on Gold Band labels. These were produced between 1917 and 1923. Just enter them under the usual label name Columbia.
"Grafonola" was also the name of a line of Columbia gramophones, that's true. To get away from the "ugly" outside horns on gramophones and make their machines more look like unobtrusive pieces of elegant furniture, Victor introduced machines with internal horns in 1906 and called them Victrolas. Competitors like Columbia Graphophone Co. quickly followed suit by coming out with their own internal horn gramophones and calling them something ending in -ola, like Columbia's Grafonola gramophones, which were introduced in 1907: https://en.wikipedia.org/wiki/Columbia_Grafonola