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

Solution 1:

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.

Solution 2:

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