Why do acetylene and chlorine react to yield carbon and HCl, but acetylene and bromine react less intensely?
Bromine is not as strong of an oxidizing agent as chlorine.  In essence the reaction of acetylene with chlorine is the same as burning, just with elemental halogen instead of elemental oxygen.  

The higher electronegativity of chlorine explains why it is a good oxidant as compared to bromine.

Other notes:  


Describing something as 'radically' (this phrasing removed  from original question edit) when you mean 'intensely' will
cause much confusion around here (Stack) since 'radically' refers to reactions involving single electron mechanisms.