Many language academies don't explain the difference between these two forms of expressing obligation sufficiently. At Advance English Igualada, we are experts in precisely and concisely clarifying such grammatical structures.
While both express obligation, there are clear differences between them as the origin of obligation can also differ. Of the two, "Have to" is the strongest given that this expresses an imposed obligation which leaves you with no choice other than to comply with the obligation. Think about Taxation, this is a legal obligation that leaves us with no choice: "I have to pay tax".
"Must" however tells us more about a self-imposed obligation, something we do because we know this will somehow benefit us: "I must change my lifestyle as I'm unfit and overweight".
Sometimes we can use both in the same context, usually when we talk about rules and regulations in organisations such as schools or companies.
Here's a summary:
HAVE TO
Imposed obligation
No choice
Law
Consequences for not complying
MUST
Self-imposed obligation
Morale compass / Right or wrong
Beneficial to comply
BOTH
Rules
Regulations
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