I'd say there's very little difference when you're speaking in the first person saying "I need to" "I have to" or "I must."

When speaking in the second or third person there might be some subtle differences. For example "you need to do this" is more like "you should do it" and "you have to" is similar to "you must."

There are also differences in usage, with "must" being much more formal.

It's the same with Russian words надо, нужно, необходимо, должен, вынужден, следует, приходится, стоит, and all the others. They all have very subtle differences between them and it's impossible to give an exact correspondence with English verbs like need, have to, must, should, ought to.