The debate around vaccines has been going on for decades, but the COVID-19 pandemic showed how crucial it is to understand and mitigate anti-vaccine sentiments. While the pandemic may be over, it is still important to understand how the pandemic affected the anti-vaccine discourse, and whether the arguments against non-COVID vaccines (e.g., Flu, MMR, IPV, HPV vaccines) have also changed due to the pandemic. This study attempts to answer these questions through a large-scale study of anti-vaccine posts on Twitter. Almost all prior works that utilized social media to understand anti-vaccine opinions considered only the three broad stances of Anti-Vax, Pro-Vax, and Neutral. There has not been any effort to identify the specific reasons/concerns behind the anti-vax sentiments (e.g., side-effects, conspiracy theories, political reasons) on social media at scale. In this work, we propose two novel methods for classifying tweets into 11 different anti-vax concerns -- a discriminative approach (entailment-based) and a generative approach (based on instruction tuning of LLMs) -- which outperform several strong baselines. We then apply this classifier on anti-vaccine tweets posted over a 5-year period (Jan 2018 - Jan 2023) to understand how the COVID-19 pandemic has impacted the anti-vaccine concerns among the masses. We find that the pandemic has made the anti-vaccine discourse far more complex than in the pre-COVID times, and increased the variety of concerns being voiced. Alarmingly, we find that concerns about COVID vaccines are now being projected onto the non-COVID vaccines, thus making more people hesitant in taking vaccines in the post-COVID era.