In late 2021, the Omicron SARS-CoV-2 variant overtook the previously dominant Delta variant, but the extent to which this transition was driven by immune evasion or a change in the inherent transmissibility is currently unclear. We estimate SARS-CoV-2 transmission within Danish households during December 2021. Among 26,675 households (8,568 with the Omicron VOC), we identified 14,140 secondary infections within a 1-7-day follow-up period. The secondary attack rate was 29% and 21% in households infected with Omicron and Delta, respectively. For Omicron, the odds of infection were 1.10 (95%-CI: 1.00-1.21) times higher for unvaccinated, 2.38 (95%-CI: 2.23-2.54) times higher for fully vaccinated and 3.20 (95%-CI: 2.67-3.83) times higher for booster-vaccinated contacts compared to Delta. We conclude that the transition from Delta to Omicron VOC was primarily driven by immune evasiveness and to a lesser extent an inherent increase in the basic transmissibility of the Omicron variant.