"I have to" vs. "I must"

Have to contrasts with must in that it usually expresses an obligation imposed by someone other than the speaker. Had to is the past tense of have to and may be used in cases where a past equivalent of must is required. Need to is used where there is not such a strong obligation, but where completing the action will satisfy a particular requirement.

I'd like to add an important point re: the use of must expressing personal obligation.

The authors of the Longman Grammar of Spoken and Written English argue that

the personal meaning (personal obligation) [of the modal must - Alex B.] is most common in academic prose.

They also add that

the rarity of must for obligation in conversation is probably due to the strong impression must makes when used in face-to-face interaction. Should and have to are less threatening ways to express obligation in conversation.

cf. Swan 2005 361:4 or Martin Parrott's remark that "many people don't make this distinction" [have to for external obligation and must for internal obligation - Alex B.] (Parrott 2000: 125), see also Quirk et al. 1985: 226 note b.