During solar flares, a large fraction of the released magnetic energy is carried by energetic electrons that transfer and deposit energy in the Sun's atmosphere. Electron transport is often approximated by a cold thick-target model, assuming that electron energy is much larger than the temperature of the ambient plasma, and electron energy evolution is modeled as a systematic loss. Using kinetic modeling of electrons, we reevaluate the transport and deposition of flare energy. Using a full collisional warm-target model (WTM), we account for electron thermalization and for the properties of the ambient coronal plasma such as its number density, temperature and spatial extent. We show that the deposition of nonthermal electron energy in the lower atmosphere is highly dependent on the properties of the flaring coronal plasma. In general, thermalization and a reduced WTM energy loss rate leads to an increase of nonthermal energy transferred to the chromosphere, and the deposition of nonthermal energy at greater depths. The simulations show that energy is deposited in the lower atmosphere initially by high-energy nonthermal electrons, and later by lower energy nonthermal electrons that partially or fully thermalize in the corona, over timescales of seconds, unaccounted for in previous studies. This delayed heating may act as a diagnostic of both the injected nonthermal electron distribution and the coronal plasma, vital for constraining flare energetics.