From Latin 'divertere' (to turn aside) — amusement 'diverts' because it turns the mind away from its worries.
To cause something to change course or turn from one direction to another; to distract someone's attention; to entertain or amuse.
From Latin 'dīvertere' (to turn aside, to go different ways, to separate), compounded from 'dī-' / 'dis-' (apart, from PIE *dwis-) + 'vertere' (to turn). 'Vertere' derives from PIE *wert- (to turn, to wind, to become), a root widely shared across Indo-European: Sanskrit 'vartate' (turns), Avestan 'varəta' (turned), Lithuanian 'versti' (to overturn), Old Norse 'verða' (to become), German 'werden' (to become). In Classical Latin 'dīvertere' meant to turn away, to separate (roads or paths dividing), and
The French 'divertissement' — a light entertainment or interlude — preserves the original connection between turning aside and amusement. When Blaise Pascal wrote about 'divertissement' in his Pensées, he used the word in its full etymological force: entertainment is what 'turns us aside' from contemplating our own mortality. For Pascal, all human amusement is literally a diversion.