Hi Robbie,  In the 'tabatière' format [snuffbox style movement] the
axis of the barillet [spring-barrel] is perpendicular to the axis of
the pinned cylinder; in the 'cartel' format, the axis of the barillet
and the pinned cylinder are parallel.  This difference of mechanical
design explains the two terms.

That's my version!

Best regards,
Jean-Marc Lebout

 [ Thus the tabatière movement is wound by a key which turns the
 [ vertical arbor of the spring, whereas the cartel movement is wound
 [ by a horizontal winding key shaft extending from the spring barrel,
 [ or more probably by a ratchet lever ("lever wind").  -- Robbie