There are differences between the types of damage asbestos does to the lungs and that caused by cigarette smoke.
Asbestos diseases cause scarring which can be seen on x-ray examinations and eventually restricts the ability of our lungs to inhale as much air, which is called a restrictive airway change. Cigarette smoke increases inflammation and mucus in the airways which obstruct airway movement, which is called an obstructive change.
Lung cancer has been shown to be caused by both tobacco smoke and asbestos exposure, as well as other carcinogens. Exposure to both tobacco smoke and asbestos greatly enhances that risk so that the total risk is greater than adding the individual effects (a toxicology effect called synergism). If asbestos exposure increases your chance of getting cancer by 5 times and smoking increases your chance of getting cancer by 12 times, then being exposed to both of them can increase your chances by 60 to 100 times.
There is evidence that quitting smoking will reduce the risk of lung cancer among people who have been exposed to asbestos dust, perhaps by as much as half after at least 5 years without smoking tobacco.