A strong cigar (e.g. Montecristo #2, Partagas Shorts) refers to the nicotine “punch” of the cigar — a strong cigar has a high nicotine content and contributes to the “light headedness” a smoker might experience. This feeling can be either quite pleasurable or, if too strong, lead to a bout of nausea. Most experienced smokers know their own limits, but a junior smoker who jumps directly into the strong smokes e.g. Partagas Shorts, Montecristo #2 may experience some discomfort.
A full bodied cigar (e.g. Bahia Maduro robusto) refers to the “bitterness” of the cigar. A milder cigar is often very floral (tastes like perfume?), or cedary (woody), while a full bodied cigar tastes like very bitter dark chocolate: creamy and pervading on the palate.
Whether or not a cigar is strong or full bodied is often open to much debate since each smoker will experience a particular cigar in different ways due the relative differences of each smoker. Some factors to consider: body weight, experience in smoking cigars, the meal taken beforehand, the drink at hand during the smoke, the speed at which the cigar is smoked, etc.
With these factors to consider, its even possible for the same smoker to smoke the same cigar at different times and have 2 completely different experiences!