anger
From the Old Norse angr, meaning "distress, grief," from Proto-Germanic angus, ultimately from Proto-Indo-European *angh, "stretch round, tight, painfully constricted, painful."
angst
From Old High German angust, from the same root as anger.
anguish
From Old French anguisse, meaning "choking sensation," from Latin angustia, "tightness, distress," from Latin anguere, "to throttle, torment," ultimately from Proto-Indo-European *angh.
anxiety
From the Latin anxietatem, noun of quality from anxius, "solicitous, uneasy, troubled in mind," also from anguere.
These four words all refer to intense negative emotions, and they're all different, but once you see the roots, it's easy to see how they could have differentiated and developed specialized senses.