Thermally driven escape from planetary atmospheres changes in nature from an organized outflow (hydrodynamic escape) to escape on a molecule-by-molecule basis (Jeans escape) with increasing Jeans parameter, lambda, the ratio of the gravitational to thermal energy of the atmospheric molecules. This change is described here for the first time using the direct simulation Monte Carlo method. When heating is predominantly below the lower boundary of the simulation region, R-0, and well below the exobase of a single-component atmosphere, the nature of the escape process changes over a surprisingly narrow range of Jeans parameters, lambda(0), evaluated at R-0. For an atomic gas, the transition occurs over lambda(0) similar to 2-3, where the lower bound, lambda(0) similar to 2.1, corresponds to the upper limit for isentropic, supersonic outflow. For lambda(0) > 3 escape occurs on a molecule-by-molecule basis and we show that, contrary to earlier suggestions, for lambda(0) > similar to 6 the escape rate does not deviate significantly from the familiar Jeans rate. In a gas composed of diatomic molecules, the transition shifts to lambda(0) similar to 2.4-3.6 and at lambda(0) > similar to 4 the escape rate increases a few tens of percent over that for the monatomic gas. Scaling by the Jeans parameter and the Knudsen number, these results can be applied to thermally induced escape of the major species from solar and extrasolar planets.