SARS-CoV-2 is thought to have emerged from bats, possibly via a secondary host. Here, we investigate the relationship of spike (S) glycoprotein from SARS-CoV-2 with the S protein of a closely related bat virus, RaTG13. We determined cryo-EM structures for RaTG13 S and for both furin-cleaved and uncleaved SARS-CoV-2 S; we compared these with recently reported structures for uncleaved SARS-CoV-2 S. We also biochemically characterized their relative stabilities and affinities for the SARS-CoV-2 receptor ACE2. Although the overall structures of human and bat virus S proteins are similar, there are key differences in their properties, including a more stable precleavage form of human S and about 1,000-fold tighter binding of SARS-CoV-2 to human receptor. These observations suggest that cleavage at the furin-cleavage site decreases the overall stability of SARS-CoV-2 S and facilitates the adoption of the open conformation that is required for S to bind to the ACE2 receptor.