The advancements in the development of Large Language Models have evolved as a transformative paradigm in conversational tasks which has led to its integration in the critical domain of healthcare. With LLMs becoming widely popular and their public access through open-source models, there is a need to investigate their potential and limitations. One such critical task where LLMs are applied but require a deeper understanding is that of self-diagnosis of medical conditions in the interest of public health. The widespread integration of Gemini with Google search, GPT-4.0 with Bing search, has led to shift in trend of self-diagnosis from search engine LLMs. In this paper, we prepare a prompt engineered dataset of 10000 samples and test the performance on the general task of self-diagnosis. We compare the performance of GPT-4.0 and Gemini model on the task of self-diagnosis and record accuracies of 63.07% and 6.01% respectively. We also discuss the challenges, limitations, and potential of both Gemini and GPT-4.0 for the task of self-diagnosis to facilitate future research and towards the broader impact of general public knowledge. Furthermore, we demonstrate the potential and improvement in performance for the task of self-diagnosis using Retrieval Augmented Generation.