Rural areas in Tibet, with its complex terrain, fragile ecology, and poor facilities, are subject to a combination of social–ecological system elements, facing the typical risks of recurrent, marginal, and potential poverty. At present, the spatial differentiation and driving mechanism of rural spatial poverty risk in Tibet are not clear, which adversely affects the formulation of differentiated and precise governance strategies. Thus, based on the social–ecological system perspective, 967 poor rural villages in eastern Tibet were taken as an example, using intelligent techniques such as random forest, geographic detector, and multi-scale geographically weighted regression to identify the spatial differentiation characteristics and the driving mechanism of poverty. The results indicated that (1) the high poverty incidence of rural areas in eastern Tibet showed a scattered block distribution, of which approximately 37% of the villages presented a spatial distribution characterised by a high degree of clustering of the high poverty incidence. (2) Topography and the level of public facilities were key factors influencing the poverty levels of rural areas in eastern Tibet, in which the coupling explanatory power between the construction land slope index (CLSI) and several poverty-causing factors was high. (3) Geological disaster, land surface temperature, CLSI, traffic accessibility, livestock resources, cropland per capita, and tourism resources differentially drove the poverty incidence of rural areas in eastern Tibet, forming spatial partitions dominated by the risks of potential, marginal, and recurrent poverty. For different partitions, differentiated governance strategies of upgrading ecological environments, optimising geographical locations, and revitalising social resources were proposed to provide references for solving the problem of relative poverty in the new period.