What is the difference between have to must and need to?

According to Cambridge University Press, they all express necessity and are similar in meaning, thus interchangeable. In colloquial speaking they are more or less used in the same way. According to Oxford, "need" is the personal opinion of what is necessary, "have to" expresses a general obligation outside the control of the speaker, as in a law or rule, and "need to" is used in both contexts. However, I have read in many places that need and have to are not officially modals since they change in tense and add third person -s. I am more likely to agree with the Cambridge source, personally.