German to English dictionaries define both words as meaning 'violence'. As far as I can understand from Wiktionary, 'Gewalt' refers to the concept of violence, while 'Gewalttätigkeit' refers to the use of violence. Is this interpretation correct, or are they interchangeable?
Gewalttätigkeit is a noun built from an adjective (like redness is built from red). That adjective is gewalttätig (violent), and it is used to describe a person or a group who is committing violent acts or is prone to committing violent acts ("Gewalttaten").
So other than Gewalt, Gewalttätigkeit can be used as a trait of a person or a group. The English language uses "violence" for both afaik, but you can maybe also use something like "violent nature" or "proneness to violence" for Gewalttätigkeit.
Die Gewalttätigkeit der haitianischen Geheimpolizei war überall im Land bekannt und gefürchtet. (The violence of the Haitian secret police was known and feared across the country.)
Auffallend sind die außergewöhnliche Empathielosigkeit und Gewalttätigkeit des Angeklagten. (What is striking is the exceptional lack of empathy and proneness to violence on the part of the accused.)
Für alle Beobachter schockierend war die Gewalttätigkeit der Polizisten bei der Auflösung der Proteste (The violence of the policemen when the protests were broken up was shocking to all observers.).
In all three examples, Gewalttätigkeit does not directly refer to the acts of violence themselves, it is ascribed to the persons. The English translations are more ambiguous in that respect.
"Gewalt" more directly refers to the violent act or to the concept of violence:
"Und bist du nicht willig, so brauch ich Gewalt!" ("And if you are not willing, I will use force!": Familiar quotation of a threat from a Goethe ballad)
Keine Gewalt! ("No violence!": Chant of GDR protestors against "Stasi" secret police forces on the street in 1989.)
Gewalt gegen Kinder steht unter Strafe. (Violence against children is a punishable offence.)
Gewalttätig is literally the combination of "Gewalt" (=violence) and "Tätigen" (=doing a deed, deed = Tat) so it's the use of violence while "Gewalt" usually usually refers to what is being done. Also "Gewalt" might in other contexts also be used as a synonym for control or power like "separation of powers" would be "Gewaltenteilung" in German while you'd use gewalttätig more for brutish violence and usually only as an adjective an rarely as a noun.
"Gewalt" has a slightly different meaning in German than "violence" in English. It means "a force which is difficult to deal with". That can be physical, than it is "violence" as has already been mentioned, but also "force". "Gewalt" comes from the (practically extinct, save for compounds and collocations) Verb "walten" which itself is derived from the Old-High-German "waltan" meaning to rule or to exercise. The English word derived from the same root is "to wield".
It is true, "Gewalt" on itself is (usually) translated with "violence" if physical force is used, but:
Gewalt = violence
mit Gewalt = using brute force (also: to force sth. against all odds, make something fit that doesn't fit at all, etc.)
Naturgewalt = force/power of nature
gewaltsam = either "violently" or like "mit Gewalt"
gewaltig = tremendous, vast, enormous but also: daunting
"Gewalttätig(-keit)" is the trait of using raw power or raw force ("Gewalt") to achieve a goal and is almost exclusively used in a physical sense. The suffix "-keit" is just a marker for a an Adjektiv/Adverb made noun, like "-ness" or "-ty" in English.
The Adjektiv "gewalttätig" is derived from "Gewalt" itself and "tätig", which comes ultimately from "tun" (to do). "Tat" is something one does (deed, act), "tätig [sein]" means about "to be doing" or "to work".