1) He smoked fewer cigarettes than me.
This refers to the quantity of cigarettes, e.g., if on a particular day, he smoked 5, but you smoked 10.
Also, for this reason - that of us discussing a quantity of items, as opposed to something singular - the correct word is "fewer", although "less" can be just as, if not more(!), idiomatic.
2) He smoked cigarettes less than me.
This refers more to the frequency of how often you smoke. It's more general, and isn't referring directly the number of cigarettes, but instead to how often you generally smoke cigarettes. It refers to "smoking cigarettes" as an activity.
3) He smoked less than me.
This is the same as 2), except that you are not specifying what you are smoking, but just talking about smoking in general.