A dirty chai is a chai latte with a shot of espresso added. A chai latte is spiced black tea and steamed milk, sweet and warming; the espresso in a dirty chai adds coffee flavour and a bigger caffeine kick.
The differences
Coffee: a dirty chai has a shot of espresso; a chai latte has none.
Caffeine: about 40–50 mg in a chai latte, roughly 100 mg or more in a dirty chai.
Taste: a dirty chai is spicier and more bitter-edged; a chai latte sweeter and softer.