I play the C-Griff, but B-Griff and C-Griff are both equally suited for the chromatic button accordion.
For what I have heard from accordion teachers, both systems b-system and c-system can do it all.
With b-system the angle of the wrist of the right hand is different from c-system. In C-Griff the arm and wrist are more in a straighter line.
I have no proof or historic documents about this, but I think the B-Griff was patented first in the area of Vienna.
Some name Franz Walter of Vienna, around 1850.
The oldest surviving example of a B-Griff Schrammelharmonika is from 1873 or 1874, built by Walter. This is written in the thesis by Andreas Teufel (Magisterarbeit Die Schrammelharmonika, a 26 Megabyte document free for download). There are 3 little photos of this 1873 instrument in this thesis. It is said to be owned by an accordionist from Graz in Austria.
Another example is in the Brussels MIM Museum, the museum of musical instruments, dated 1888 , built by Karl Budowitz. This photo can be found on the MIM website. I have seen it a week ago, when I went to the MIM in Brussels for the 200 years Adolphe Sax and saxophone exhibition.
It is possible another maker changed the 1st and 3rd row reed blocks, to avoid patents for the B-Griff, and so created a C-Griff. Some name Mirwald in 1891 in Bavaria, Germany (Alfred Mirek). Others name Paolo Soprani, Mattia Berardi, Piatanesi, who patented in 1897 a C-Griff with standard basses.
Some sources say the B-Griff was developed in Vienna around 1860-1870-1880, others already in 1850 by Franz Walter. Still unclear.
I think the 2 different systems C-system and B-system developed to avoid trouble with patents. But this is only a wild guess.