Because of the spread of the Omicron variant, many countries have experienced COVID-19 case numbers unseen since the start of the pandemic. We aimed to compare the epidemiological characteristics of Omicron with previous variants and different strains of influenza to provide context for public health responses. We developed transmission models for SARS-CoV-2 variants and influenza, in which transmission, death, and vaccination rates were taken to be time-varying. We fit our model based on publicly available data in South Africa, the United States, and Canada. We used this model to evaluate the relative transmissibility and mortality of Omicron compared with previous variants and influenza. We found that Omicron is more transmissible and less fatal than both seasonal and 2009 H1N1 influenza and the Delta variant; these characteristics make Omicron epidemiologically more similar to influenza than it is to Delta. We estimate that as of February 7, 2022, booster doses have prevented $4.29\times {10}^7$ and $1.14\times {10}^6$ Omicron infections in the United States and Canada, respectively. Our findings indicate that the high infectivity of Omicron will keep COVID-19 endemic, similar to influenza. However, because of Omicron's lower fatality rate, our work suggests that human populations living with SARS-CoV-2 are most likely.