The emergence of several SARS-CoV-2 variants has caused global concerns about increased transmissibility, increased pathogenicity, and decreased efficacy of medical countermeasures. Animal models can be used to assess phenotypical changes in the absence of confounding factors that affect observed pathogenicity and transmissibility data in the human population. Here, we studied the pathogenicity of variants of concern (VOC) B.1.1.7 and B.1.351 in rhesus macaques and compared it to a recent clade B.1 SARS-CoV-2 isolate containing the D614G substitution in the spike protein. The B.1.1.7 VOC behaved similarly to the D614G with respect to clinical disease, virus shedding and virus replication in the respiratory tract. Inoculation with the B.1.351 isolate resulted in lower clinical scores in rhesus macaques that correlated with lower virus titers in the lungs, less severe histologic lung lesions and less viral antigen detected in the lungs. We observed differences in the local innate immune response to infection. In bronchoalveolar lavages, cytokines and chemokines were upregulated on day 4 in animals inoculated with D614G and B.1.1.7 but not in those inoculated with B.1.351. In nasal samples, we did not detect upregulation of cytokines and chemokines in D614G or B.1.351-inoculated animals. However, cytokines and chemokines were upregulated in the noses of B.1.1.7-inoculated animals. Taken together, our comparative pathogenicity study suggests that ongoing circulation under diverse evolutionary pressure favors transmissibility and immune evasion rather than an increase in intrinsic pathogenicity.