Healthcare workers such as doctors and nurses are expected to be trustworthy and creditable sources of vaccine-related information. Their opinions toward the COVID-19 vaccines may influence the vaccine uptake among the general population. However, vaccine hesitancy is still an important issue even among the healthcare workers. Therefore, it is critical to understand their opinions to help reduce the level of vaccine hesitancy. There have been studies examining healthcare workers' viewpoints on COVID-19 vaccines using questionnaires. Reportedly, a considerably higher proportion of vaccine hesitancy is observed among nurses, compared to doctors. We intend to verify and study this phenomenon at a much larger scale and in fine grain using social media data, which has been effectively and efficiently leveraged by researchers to address real-world issues during the COVID-19 pandemic. More specifically, we use a keyword search to identify healthcare workers and further classify them into doctors and nurses from the profile descriptions of the corresponding Twitter users. Moreover, we apply a transformer-based language model to remove irrelevant tweets. Sentiment analysis and topic modeling are employed to analyze and compare the sentiment and thematic differences in the tweets posted by doctors and nurses. We find that doctors are overall more positive toward the COVID-19 vaccines. The focuses of doctors and nurses when they discuss vaccines in a negative way are in general different. Doctors are more concerned with the effectiveness of the vaccines over newer variants while nurses pay more attention to the potential side effects on children. Therefore, we suggest that more customized strategies should be deployed when communicating with different groups of healthcare workers.