There sure is difference between the two. As a rule the context shows if it is of importance. Say, if you don't want to do something but have little choice, you say: 'I have to do it.'

More options:

• If you are eager to do something but see or foresee some obstacles, you say: 'I have got to/gotta do it' ('I always knew I had got to do it' [Tina Turner in an interview]);
• If the necessity is inflicted by circumstances and your will is of little importance (agreements, plans, schedules, curricula, &c.), you say: 'I am to do it'. ('I am to do the talking and you are to do the listening' ['Farewell My Darling' by Raymond Chandler]).

'Must' doesn't have the distinct connotation of, so to say, 'inevitability' 'have to' has, if you ask me.

Your examples fit in as well, by the way.

Yes they are similar, but...

When you want to say that it is required something to be done regularly you use have to

He has to do his homework after school every day.

Formally, when you speak about the law or obligation you use must

Everyone must pay taxes.

Must has no past form and no future

For the past obligation you use, for example, had to

She had to go to work.

For the future obligation you use, for example, will have to

We will have to manage that properly.

Negatives are seriously different

I don't have to go to school = there is no need for me to go to school

I must not go to school = it is forbidden for me to go to school

If you believe in certain fact you use must

You must be that man from the TV.

(If you believe something is not true you say You can't be the man from the TV.)

You use have to with the similar meaning but not with you

It has to be true.

(negative: It can’t be true.)

If you believe that some process is in order you use must+be+ing not have to

She is not at home, she must be working

She has to be working is simply She has an obligation to be working

For a particular situation resolution or obligation you say have got to

I've got to report this accident.

The differences are not crucial, but are important.

(There is also need to with a few peculiarities on its own, and it can express an obligation or an order as well.)