Recently, there has been a growing trend of utilizing Large Language Model (LLM) to evaluate the quality of other LLMs. Many studies have fine-tuned judge models based on open-source LLMs for evaluation. While the fine-tuned judge models are claimed to achieve comparable evaluation capability with GPT-4, in this work, we conduct an empirical study of LLM-as-a-Judge. Our findings indicate that although the fine-tuned judge models achieve high performance on in-domain test sets, even surpassing GPT-4, they underperform GPT-4 across several dimensions, including generalizability, fairness and adaptability. We also reveal that the fine-tuned judge model inherently operates as a task-specific classifier, consequently imposing the limitations.