They would both appear to mean the same thing in English i.e. 'still'
How do they differ and under what circumstances might you use either of them?
"Noch immer" is used in cases where the speaker has already expressed in the past their wish or assumption that the described situation would have ended by the time of speaking - which has not happened.
It is a stronger and more emotional form of "noch", as can be seen for example in the translation of Edgar Allan Poe's "The Raven":
I noticed "noch immer" is quite often being used to describe activities or a state of something that is regrettably not changed yet.
"Noch" has no such connotation, it is completely neutral.