The ubiquity and widespread use of digital and online technologies have transformed mental health support, with online mental health communities (OMHCs) providing safe spaces for peer support. More recently, generative AI and large language models (LLMs) have introduced new possibilities for scalable, around-the-clock mental health assistance that could potentially augment and supplement the capabilities of OMHCs. Although genAI shows promise in delivering immediate and personalized responses, its effectiveness in replicating the nuanced, experience-based support of human peers remains an open question. In this study, we harnessed 24,114 posts and 138,758 online community (OC) responses from 55 OMHCs on Reddit. We prompted several state-of-the-art LLMs (GPT-4-Turbo, Llama-3, and Mistral-7B) with these posts, and compared their responses to human-written (OC) responses based on a variety of linguistic measures across psycholinguistics and lexico-semantics. Our findings revealed that AI responses are more verbose, readable, and analytically structured, but lack linguistic diversity and personal narratives inherent in human--human interactions. Through a qualitative examination, we found validation as well as complementary insights into the nature of AI responses, such as its neutral stance and the absence of seeking back-and-forth clarifications. We discuss the ethical and practical implications of integrating generative AI into OMHCs, advocating for frameworks that balance AI's scalability and timeliness with the irreplaceable authenticity, social interactiveness, and expertise of human connections that form the ethos of online support communities.