Understanding the dynamics of counseling conversations is an important task, yet it is a challenging NLP problem regardless of the recent advance of Transformer-based pre-trained language models. This paper proposes a systematic approach to examine the efficacy of domain knowledge and large language models (LLMs) in better representing conversations between a crisis counselor and a help seeker. We empirically show that state-of-the-art language models such as Transformer-based models and GPT models fail to predict the conversation outcome. To provide richer context to conversations, we incorporate human-annotated domain knowledge and LLM-generated features; simple integration of domain knowledge and LLM features improves the model performance by approximately 15%. We argue that both domain knowledge and LLM-generated features can be exploited to better characterize counseling conversations when they are used as an additional context to conversations.