What is the difference between 抽烟 (chōu yān) and 吸烟 (xī yān) which both pertain to smoking?
Basically they have the same meaning and could be used interchangeably. 

The subtle difference is, 抽烟 is used in spoken language more than 吸烟. As you said, when asking people we usually say 你抽烟吗; of course 你吸烟吗 is correct too. 

On the other hand, 吸烟 is more formal and used in written language more. As you've seen, for a slogon 吸烟 is used usually; 禁止抽烟 is not wrong, but not used in general.