Smoking is not allowed here is a definite and final way to say any act of smoking must not be carried out on that premises. The 'not' in the sentence emphasizes that any breach of the anti-smoking law in that vicinity will be legally sanctioned. On the other hand, 'You may not smoke here' is more relaxed and left to the discretion of the smoker to decide if to smoke or not to smoke in the vicinity. The law on smoking in such area is relaxed and the smoker can as a matter of courtesy decide not to smoke even if he won't be sanctioned for smoking. The 'may' in the sentence denotes probability of something that can happen either way and it is up to the smoker to decide on what to do in such environment.

"Not allowed" is more like a rule that has put down by someone else. "May not smoke here" is like someone just telling you that you can't smoke in that place. I am sure there is better explaining for it, that's just how I see the difference between the two.

They are essentially the same thing, the tone is just a bit different. "Smoking is not allowed here" sounds like a rule or warning you'd see on a sign while "you may not smoke here" means the same thing but said in a nicer tone. You could word it in a lot of ways, but the tone can dictate how it's interpreted. From most to least strict: Smoking within this area is strictly forbidden by law. You are not allowed to smoke here. No smoking allowed. You may not smoke here. Please do not smoke here.