Everybody who stops smoking will notice the difference immediately. Your physical condition improves, you have a fresh breath again and you smell better. The nose starts to work better (you smell more) and food tastes better. After a while, smoker's cough will disappear as well. The thing that you don't notice is even more important: your blood pressure goes down, heart beat goes down to a normal level, the whole body starts to function better again.
The time it takes before the health risks of the ex-smoker are equal to those of someone who has never smoked depends on the amount and time that someone has smoked. After a couple of years, the increased risk of heart and lung diseases is reduced considerably. The first years after quitting show the biggest decrease of these risks. And after 10 to 15 years, the ex-smoker is as healthy again as someone who has never smoked. Within several years, the risk of heart and coronary diseases are equal to that of non-smokers. Also, the risk of lung disease decreases relatively quickly. The risk of lung cancer needs 10 to 15 years to decrease to the risk of someone who never smoked. But also in this case the biggest decrease of the risk happens in the first years after quitting.
There are people who die of a smoker's disease shortly after they quit smoking. This is, of course, not because quitting is unhealthy, but because they already had the disease. A tumour in the lungs, for example, could have been there unnoticed for a long time.