I'm sure that you already know how to use 'not' to make the verb in a clause or sentence negative, so I'll give you some examples of how 'non' is used. We use 'non' in combination with other words - usually nouns or adjectives - to form compound nouns and adjectives, often with a hyphen (-). We generally use these compound words to show a contrast of some kind. Here are some examples: Books and films that are 'fiction' are based on invented stories, while books that are 'non-fiction' are based on facts. Something which is 'toxic' is poisonous, while something which is 'non-toxic' isn't poisonous. A hotel might distinguish between 'residents' (people staying as guests at the hotel) and 'non-residents' (people who are not staying). A product might be made from recyclable material (if it can be recycled) or non-recylable material (if it can't). A drink or other food product might be made from dairy products (cows' milk, cheese etc) or non-dairy products (soya, almond milk etc). In your example, the correct phrase is 'non-smoking area'. The compound adjective 'non-smoking' describes an area where smoking is not allowed, as opposed to the 'smoking' area where smoking is allowed.

Not is more commonly used ie Do not smoke whereas on signs and things you will most likely see non ie Non smoking area

Not smoking area