Have to and Must

Have to and must are being looked at together because of the inter-changeability when used for certain functions and the confusion caused when they cannot be interchanged for others.

Have to

'Have to' is used to express certainty, necessity, and obligation.
This has to be the right place. We are not lost. Certainty
The glue has to be left to dry for 24 hours. Necessity
I have to leave early. Obligation

'Have to' changes when it is used in the present, past or future.
Here are some examples of its functions and forms:

Certainty

Present That has to be Peter calling.
Negative That can't be Peter calling. He doesn't have my number. For a negative 'must not' or 'cannot' are used.

Past That has to have been Peter calling.
Negative That cant have been Peter calling. He doesn't have my number.

With regard certainty in the future other verb forms are used.

Necessity

Present We have to finish this today./ We didn't have to finish that yesterday.
Past We had to finish that yesterday. / We didn't have to finish that yesterday.
Future We will have to finish this tomorrow. / We won't have to finish

In the case of necessity 'must' can also be used. However in the negative 'do not have to' means that it is not necessary but ‘must not’ means it is forbidden to do something.

We do not have to finish this today. – It is not necessary.
We must not finish this today. – It is forbidden.

Must

'Must' is mostly used to express certainty however as seen above it can also be used for necessity. It is also used to express strong obligation. Most native speakers prefer ‘have to’ which is less restrictive in its meaning. For prohibition 'Must not' can be used but the meaning is extremely severe and native speakers prefer 'should not' or 'ought not to' which gives the idea of dissuasion rather than prohibition.

Certainty This must be the right address.
Necessity All passengers must present their passports at the gate.
Strong recommendation You really must see someone about that cold.
Prohibition Max, you must not run in the street.

Here are some examples of 'must' in its forms and functions. Notice how it changes depending on present, past and future time.

Certainty

Present That must be Peter calling. / That mustn't be Peter calling.
Past That must have been Peter calling. / That mustn't have been Peter calling.

For negative certainty most native speakers prefer 'cannot' to 'must not'
For positive certainty 'have to' is also used.

Prohibition

You must not play music after eleven o'clock.
You must not forget to clean the equipment after you use it.

Prohibition usually refers to the near future.

Recommendation

Present You must go to see a doctor. You must not smoke so.
Past (should) You should have gone to see a doctor. You should not drink so much tomorrow.

Necessity

Present You must show an I.D. to enter. / You don't have to show an I.D. to enter.
Past You had to show an I.D. to enter. / You didn't have to show an I.D. to enter.
Future You must show your I.D. to enter tomorrow. / You will not have to show an I.D. to enter.

For the negative of 'must' for necessity 'have to' is used.