Mining and analysis of the Big Data of Twitter conversations have been of significant interest to the scientific community in the fields of healthcare, epidemiology, big data, data science, computer science, and their related areas, as can be seen from several works in the last few years that focused on sentiment analysis and other forms of text analysis of Tweets related to Ebola, E-Coli, Dengue, Human papillomavirus (HPV), Middle East Respiratory Syndrome (MERS), Measles, Zika virus, H1N1, influenza-like illness, swine flu, flu, Cholera, Listeriosis, cancer, Liver Disease, Inflammatory Bowel Disease, kidney disease, lupus, Parkinson's, Diphtheria, and West Nile virus. The recent outbreaks of COVID-19 and MPox have served as "catalysts" for Twitter usage related to seeking and sharing information, views, opinions, and sentiments involving both these viruses. While there have been a few works published in the last few months that focused on performing sentiment analysis of Tweets related to either COVID-19 or MPox, none of the prior works in this field thus far involved analysis of Tweets focusing on both COVID-19 and MPox at the same time. With an aim to address this research gap, a total of 61,862 Tweets that focused on Mpox and COVID-19 simultaneously, posted between May 7, 2022, to March 3, 2023, were studied to perform sentiment analysis and text analysis. The findings of this study are manifold. First, the results of sentiment analysis show that almost half the Tweets (the actual percentage is 46.88%) had a negative sentiment. It was followed by Tweets that had a positive sentiment (31.97%) and Tweets that had a neutral sentiment (21.14%). Second, this paper presents the top 50 hashtags that were used in these Tweets. Third, it presents the top 100 most frequently used words that are featured in these Tweets. The findings of text analysis show that some of the commonly used words involved directly referring to either or both viruses. In addition to this, the presence of words such as "Polio", "Biden", "Ukraine", "HIV", "climate", and "Ebola" in the list of the top 100 most frequent words indicate that topics of conversations on Twitter in the context of COVID-19 and MPox also included a high level of interest related to other viruses, President Biden, and Ukraine. Finally, a comprehensive comparative study that involves a comparison of this work with 49 prior works in this field is presented to uphold the scientific contributions and relevance of the same.