What is the difference between 抽烟 (chōu yān) and 吸烟 (xī yān) which both pertain to smoking? Question: What is the difference between 抽烟 (chōu yān) and 吸烟 (xī yān) which both pertain to smoking?

I‘ve ordinarily asked people


  你抽烟吗 (nǐ chōu yān ma = "do you smoke?")


for which nobody has acted oddly about (at least, in regards to my Chinese), so I assumed it was correct, but maybe


  你吸烟吗 (nǐ xī yān ma = "do you smoke?")


is synonymous.

However, I ordinarily see


  禁止吸烟 （jìn zhǐ xī yān = "smoking is forbidden")


on No Smoking signs, e.g.



which, up until recently, I thought said


  禁止抽烟 (jìn zhǐ chōu yān = "smoking is forbidden" [presumably])


but this is incorrect, and it doesn't seem to be commonly used (possibly even an invalid phrase).
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.