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Present obligation with must / have to
Despite what many grammar books tell you, there is almost no difference in meaning between must and 
have to in the positive form. Yes, many grammar sources talk about internal and external obligation, but
these are interchangeable and the strong obligation is the same.
You must / have to be over 18 to buy alcohol in this town.
I must / have to go to bed early tonight as I have an exam in the morning.
In the negative form, and with prohibition, this is where there is a very clear difference between the two.
In fact, mustn’t and don’t have to are almost opposite in their meanings. Mustn’t is a way to express 
strong prohibition, whereas don’t have to shows there is a complete lack of obligation.
You mustn’t smoke in the house. (100% do not do it!)
You don’t have to stay here if you don’t want to. (Go if you want to, but there’s no pressure to
stay or to go)
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We can’t use two modal verbs together, one after the other. Instead, we use a semi-modal.
He won’t can meet us at work in the morning. 
(Wrong)
He won’t be able to meet us at work in the morning. 
(Right)
Present obligation with should / ought to 
It is not so common these days to hear somebody using ought to, and it is a rather formal way to express
obligation. As far as differences go, ought to shows a slightly stronger obligation than should, but
generally speaking we use should much more often.
You really should / ought to go and see the doctor if you feel so ill.
I think you should / ought to go and say sorry because you upset Tim last night.