"müssen" is more like an informal word in German. It always has a simple, even childish or naive touch to it, more so when it is used without a verb, like so:
"Ich muss aufs Klo." = "I need to go to the toilet"
"Ich muss in die Schule." = "I have to go to school"
"Ich muss Hausaufgaben machen." = "I've got homework to do."
"Ich muss jetzt Fernsehen gucken." = "I (strongly, absolutely) want to watch TV now."
"Ich muss jetzt was trinken." = "I'm so thirsty."
"Ich muss nicht zur Schule gehen." = "I don't need to go to school."
(Remember this: "Ich muss nicht ..." is not "I must not" but "I don't have to")
In slightly more formal or polite contexts, we weaken "müssen" with some other construction, mostly conjunctives like:
"Ich müsste mal Ihre Toilette benutzen." = "Sir, I would need to use your toilet."
"Du solltest öfter deine Hausaufgaben machen." = "You should do your homework more often."
"Es wäre schön, wenn du das Fenster aufmachst." = "It would be nice if you open the window."
"Du brauchst nichts zu sagen." = "You don't need to say anything."
(Compare "Du musst nichts sagen." = More impolite version.)
In official contexts, much is done to avoid the word "müssen". Complicated constructions are used for this.
For example: "haben zu" is pretty much never used in informal contexts.
So, long story short: In German it's not so much about the exact meaning, but about politeness and formal or informal contexts. There are many nuances depending on what you say to whom and where and in what context, especially in case of "müssen".
A typical reaction of a German, when you use "Du musst ...", and he or she finds its usage impolite is: "Ich muss gar nichts." (= "I don't have to do (anything).")