Why are deep cycle batteries rated in amp hours instead of watts hours?
Simply put, watts measures a rate of consumption, and amp-hours or kilowatt-hours measures the total energy consumed. Batteries store energy, so a unit that quantifies the amount of energy stored is useful.

Watts are 'Joules per second', so 1 Watt is the consumption of 1 Joule per second. Watt-hours  is the energy consumed by a load with 1 Watt power consumption for one hour. This is the same unit that the power company uses to determine your electric bill.

So the question may be altered to be "Why are batteries expressed in Amp-Hours instead of KiloWatt-Hours", and there's a good reason for that as well. By specifying voltage separately, the unit of 'Amp-Hours' allows convenient recalculation for different series/parallel combinations of batteries, as well as easier calculation of battery lifespan (measuring load current is easier than load power).