Commodore produced the first generation of "Commodore 64
" until May 1986, then it was discontinued and they introduced the Commodore 64C (C64C for short). According to the 64'er magazine, this version has been planned since the Hannover mass in 1985, but as the old version sold so well during Christmas '85, its release date was pushed back.
The new model does not differ much from its predecessor, the only innovation is the flatter case, which makes the keyboard (which has off-white keys now) more ergonomic (it looks like the "Commodore 128
" case), as it is less higher than the old one. But the new case did not only have advantages: due to its low profile and additional metal screening, some of the numerous hardware expansions did not fit anymore. This was changed with the "Commodore 64G
". The 64'er staff noticed that VIC-II as well as the two CIAs have new version numbers; they didn't write which, though (these are probably just bugfixed chips).
The official name for this model was C=64C, but nevertheless the German 64'er magazine decided to call it C64-II (because the first units didn't have the new name on the label at the bottom), the badges on all the 64 C's just says Commodore 64. They pointed out that this name was only valid for the 64'er magazine, but since the 64'er was the magazine for the C64 for a long time, the name was widely accepted and so this model is mostly known as C64-II in Germany.
The C64C appeared again after (or concurrently with?) the C64G, this time with the new, short board. So, although the case might look the same and the label says Commodore 64, the boards may be completely different.