Conversational agents powered by large language models (LLM) have increasingly been utilized in the realm of mental well-being support. However, the implications and outcomes associated with their usage in such a critical field remain somewhat ambiguous and unexplored. We conducted a qualitative analysis of 120 posts, encompassing 2917 user comments, drawn from the most popular subreddit focused on mental health support applications powered by large language models (u/Replika). This exploration aimed to shed light on the advantages and potential pitfalls associated with the integration of these sophisticated models in conversational agents intended for mental health support. We found the app (Replika) beneficial in offering on-demand, non-judgmental support, boosting user confidence, and aiding self-discovery. Yet, it faced challenges in filtering harmful content, sustaining consistent communication, remembering new information, and mitigating users' overdependence. The stigma attached further risked isolating users socially. We strongly assert that future researchers and designers must thoroughly evaluate the appropriateness of employing LLMs for mental well-being support, ensuring their responsible and effective application.