SARS-CoV-2 infects host cells by binding the receptor-binding domain (RBD) of its spike protein to the receptor, ACE2. A subset of highly effective spike mutations plays critical roles in altering the conformational dynamics of spike protein. Here, we use molecular dynamics simulations to investigate how spike mutations affect the conformational dynamics of spike/ACE2 complex in the D614G, Delta (B.1.617.2) and Omicron (B.1.1.529) SARS-CoV-2 variants. We observe that the increased positive-charged mutations in the Omicron spike amplify its structural rigidity and reduce its structural flexibility. The mutations (P681R in Delta and P681H in Omicron) at the S1/S2 junction facilitate S1/S2 cleavage and aid the activation of the fusion core. We report that high structural flexibility in Delta lowers the barrier for the activation of the S2 core; however, high structural rigidity in Omicron enhances the barrier for the same. Our results also explain why Omicron requires the presence of a higher number of ACE2 to activate its fusion core than Delta.