Most current models of hot-exoplanet atmospheres assume shallow heating, a strong day-night differential heating near the top of the atmosphere. Here we investigate the effects of energy deposition at differing depths in a model tidally locked gas-giant exoplanet. We perform high-resolution atmospheric flow simulations of hot-exoplanet atmospheres forced with idealized thermal heating representative of shallow and deep heating (i.e., stellar irradiation strongly deposited at ∼10^{3} Pa and ∼10^{5} Pa pressure levels, respectively). Unlike with shallow heating, the flow with deep heating exhibits a new dynamic equilibrium state, characterized by repeated generation of giant cyclonic storms that move away westward once formed. The formation is accompanied by a burst of heightened turbulence, leading to the production of small-scale flow structures and large-scale mixing of temperature on a timescale of ∼3 planetary rotations. Significantly, while effects that could be important (e.g., coupled radiative flux and convectively excited gravity waves) are not included, over a timescale of several hundred days the simulations robustly show that the emergent thermal flux depends strongly on the heating type and is distinguishable by current observations.