According to this article, "must" and "need to" have different connotations. Is there a context in which "must" has exactly the same meaning and connotation as "need to?" By context I mean everything from a sentence describing a particular situation to different styles of speech, except for the situation in which "must" and "need to" are explicitly defined to be the same.
In general American English, "must" and "need to" can both refer to requirements, and in that sense, they are synonymous. Connotations are more difficult to explain. If a document such as a college program guide, said "You must pass 120 course hours to obtain a degree" or "You need to pass 120 course hours to obtain a degree" there would be no difference in meaning or connotation.
"Must" is probably more related with forced (no other options) while "need" is no such strong - my or someone other decision (maybe with some risk of bad consequences if I will not oblige).
At least in the US, need to is often used as a "softer" alternative to must. In this case, the implications of both are identical, but I would argue that the connotations are still slightly different. Specifically, need to is chosen as a softer or more polite alternative to must specifically because it has less strident connotations. For example, it is very common to say
I need to go to the bathroom.
Everyone knows that this actually means
if I don't find an appropriate place to relieve myself soon, we will all be very embarrassed, so I must go to the bathroom.
But we tend to want to be more circumspect about bodily functions, so we choose the more tentative-seeming need to.
Similarly, there is the imperative need to, made infamous in the movie Office Space:
When your boss uses need [you] to this way, it should probably be interpreted as [you] must. The same is true for the slightly-more-direct [you] need to, especially when used by an authority figure:
You need to clean your room = You must clean your room (or face the consequences)
when said by one's parent.
But why not just use must in that situation? Again, although the underlying meaning is you must do what I say [or else] the connotations and distancing of need to allow it to sound more like a request or suggestion than the command that it actually is.