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?
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"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":
Und der Rabe rührt' sich nimmer, sitzt noch immer, sitzt noch immer
auf der bleichen Pallas-Büste überm Türsims wie vorher;
und in seinen Augenhöhlen eines Dämons Träume schwelen,
und das Licht wirft seinen scheelen Schatten auf den Estrich schwer;
und es hebt sich aus dem Schatten auf dem Estrich dumpf und schwer
meine Seele - nimmermehr.
I noticed "noch immer" is quite often being used to describe activities or a state of something that is regrettably not changed yet.
Das Vorurteil, dass Frauen schlechter Auto fahren als Männer, besteht noch immer. (The prejudice that women are worse at driving than men still exists.)
"Noch" has no such connotation, it is completely neutral.