Must and have to both express obligation.
However, they are used differently depending on who imposes the obligation.

MUST is used:

to express an obligation imposed by the speaker (yourself or someone else)

to state a rule or legal requirement,
Example: "You must wear a seat belt." said the driver.

HAVE TO is used:

when an obligation is imposed by an external source; someone else makes it necessary.

when obeying a rule or doing your duty
Example: "We have to go through security control before boarding." (airport instructions)

How to use 'must' and 'have to':

MUST: Used when something is the rule or when the speaker (yourself or someone else) thinks it is necessary.

Examples:

• I must buy flowers for my mother.(It's her birthday. I impose the obligation on myself.)
• "You must take more exercise" says the doctor. (The doctor thinks it is necessary.)
• I must ask my secretary to book a flight for me. (It is important for me not to forget.)
• Dogs must be kept on a lead. (Written on a sign in the park.)
• All pupils must wear a uniform. (It's compulsory in this school.)
• All passengers must wear seat belts.(It's the law.)
• Cyclists must wear helmets when they are on the road. (It's the law.)
• "All essays must be handed in by Friday." said the teacher.

HAVE TO: Used when an obligation is imposed by an outside source, when obeying a rule or complying with instructions, recommendations or advice.
Examples:

• I have to buy flowers for my mother-in-law. (My husband asked me to do that.)
• I have to take more exercise. (The doctor told me it was necessary.)
• I have to call the travel agency. (My boss asked me to book a flight.)
• I have to keep my dog on a lead.(That's what the sign tells me to do.)
• As part of his job, my husband has to travel a lot.
• Do you have to wear a uniform in your school? (Is it compulsory?)
• Do I have to fasten my seat belt? (Is it an obligation?)
• Enzo has to wear a suit and tie at work. (That's part of the regulations.)
• When do we have to hand in our essays? (When are we instructed to do that?)

MUSTN'T - DON'T HAVE TO:  In the negative form the meaning changes

MUSTN'T: The speaker thinks it is important NOT to do something or it forbidden.
Examples:

• You mustn't tell George ( It is important not to tell George = don't tell George.)
• Students mustn't talk during the exam. (It is forbidden.)
• Fire fighters must not drink on duty. (It is forbidden.)
• You mustn't speak to strangers. (Parents warn their children.)
• You mustn't park here. (Only residents are allowed to park here.)

DON'T HAVE TO: It is not necessary or compulsory. It is your decision.
Examples:

• You don't have to tell George (You can tell George if you like, but it isn't necessary.)
• You don't have to wear a tie (Ties are not required but you can wear one if you like.)
• You don't have to drive me home. I'll walk. (It's not necessary. I live nearby.)
• We don't have to whisper. (It's not necessary. There's nobody else here.)

'Must' for deduction:
'Must' can also be used to show that we are certain something is true.
In this case we are making a logical deduction based on the evidence we have.

◊ Julie has had a long journey.  She must  be tired.
◊ There's no heat in this room.  You must  be cold.
◊ Pablo hasn't eaten all day.  He must  be hungry.
◊ I didn't order these books.  There must  be a mistake.
◊ The house is luxurious.       They must be rich to be able to buy a house like that!.

'Must' only exists in the present tense.
'Have to' is used for other tenses.
 Examples:

"I had to go to see the dentist yesterday."  NOT: "I have must / I musted ..."
"I will have to get up early tomorrow."  NOT "I will must ...."