Let me cite excerpts from the GeoDA-Gloassary (https://geodacenter.asu.edu/node/390)
Spatial regimes are a form of spatial heterogeneity, which implies structural differences across space. When a variable is characterized by distinct distributions (e.g., with a different mean or variance) for different geographic subregions, these subregions might point to the existence of spatial regimes.
Spatial heterogeneity exists when structural changes related to location exist in a dataset. In such cases, spatial regimes might be present, which are characterized by differing parameter values or functional forms.
Unfortunately, the difference is still not clear to me. Both explanations seem to indicate that spatial regimes as well as spatial heterogeneity refer to a process' variation of mean/variance over space.