Given the high amount of information available on social media, the paper explores the degree of vaccine hesitancy expressed in English tweets posted worldwide during two different one-month periods of time following the announcement regarding the discovery of new and highly contagious variants of COVID-19—Delta and Omicron. A total of 5,305,802 COVID-19 vaccine-related tweets have been extracted and analyzed using a transformer-based language model in order to detect tweets expressing vaccine hesitancy. The reasons behind vaccine hesitancy have been analyzed using a Latent Dirichlet Allocation approach. A comparison in terms of number of tweets and discussion topics is provided between the considered periods with the purpose of observing the differences both in quantity of tweets and the discussed discussion topics. Based on the extracted data, an increase in the proportion of hesitant tweets has been observed, from 4.31% during the period in which the Delta variant occurred to 11.22% in the Omicron case, accompanied by a diminishing in the number of reasons for not taking the vaccine, which calls into question the efficiency of the vaccination information campaigns. Considering the proposed approach, proper real-time monitoring can be conducted to better observe the evolution of the hesitant tweets and the COVID-19 vaccine hesitation reasons, allowing the decision-makers to conduct more appropriate information campaigns that better address the COVID-19 vaccine hesitancy.