What is the difference between "can't" and "mustn't" in the expressing of prohibition?

I mustn't X means that it is imperative that I not do X.

I can't X means that it is not possible for me to do X.

Therefore, mustn't is the correct answer.

However, in colloquial speech, people would use the word "can't", since it is implied in the statement that you cannot use your mobile phone without breaking the law.

Correct completions for that blank would be "may not" (the abbreviation "mayn't" is very unusual) and "shouldn't". Quick illustration:

May I use my mobile phone while driving?

No, you may not. or "No, you must never use your mobile phone while driving."

Should I use my mobile phone while driving?

No, you shouldn't.

None of the options suggested form any relation to the obvious sense of the sentence in relation to being against the law. "Mustn't" comes closest.

Another good alternative is You must disuse your mobile phone while driving. (Not only may you not start using it while driving, you must stop if you were already using it.)

Oxford Dictionaries Online says that can't is simply a contraction for can not, and the entry for can includes this definition and examples:

2 be permitted to:

• you can use the phone if you want to
• nobody could legally drink on the premises

The word is certainly used that way in conversation all the time, and sometimes in the news. But not too often. It's more common for an attention-grabbing headline to use the word can't above an article that instead uses more formal words like prevent, prohibit, has no authority, or cannot legally.

Oxford Dictionaries Online includes this usage note:

Is there any difference between can and may when used to request or express permission, as in may I ask you a few questions? or can I ask you a few questions? Many people feel that can should be reserved for expressions denoting capability, as in can you swim?, rather than for those relating to permission. May is, generally speaking, a politer and more formal way of asking for something, and is the better choice in more formal contexts.

If I had to choose a word from that list I would opt for "mustn't".

Luckily, I don't have to because I would say something completely different, i.e: "Don't use your mobile while driving. It's against the law."

"You mustn't use your mobile phone while you're driving. It's against the law."

The following is a possible scenario:

I am driving in my car with my mother sitting next to me. My mobile rings and instinctively I reach to answer it. My mother looks at me sternly (or with deep concern) and tells me: "You mustn't use your mobile."

I am like a child being told what to do by her parent, in this case her authority is greater than mine. Must is often used by individuals whose authority is greater than ours, (I am simplifying here but it's generally true) and is also used for giving emphatic advice:

"You must take more exercise. Join a gym."

However, can't is also acceptable if the situation changes:

"You can't use your mobile phone while you're driving. It's against the law."

This time I am driving in my car but I have a friend sitting next to me, and (s)he reminds me that I am breaking the law. (S)he does not have any authority over me, we are equal.

We use "can" and "cannot" to talk about having permission to do or not do something.

"I can't use my mobile" can mean I am not allowed; it is not allowed by the police/the law or it is not right thing to do.

Don't have to and needn't both mean that something is unnecessary, they have completely different meanings from can't and mustn't.

can and must

I think in the context of the original sentence,'can't' expresses that using your mobile phone while driving is not allowed, and 'mustn't' expresses necessity to not do it. In this context either can be used, but that's not always the case.

eg. Some people think that home-schooled students can't sit high school exams - 'mustn't' would not fit here.