Large language models have become increasingly common, used by millions of people worldwide in both professional and personal contexts. As these models continue to advance, they are frequently serving as virtual assistants and companions. In human interactions, effective communication typically involves two types of empathy: cognitive empathy (understanding others' thoughts and emotions) and affective empathy (emotionally sharing others' feelings). In this study, we investigated both cognitive and affective empathy across several small (SLMs) and large (LLMs) language models using standardized psychological tests. Our results revealed that LLMs consistently outperformed humans - including psychology students - on cognitive empathy tasks. However, despite their cognitive strengths, both small and large language models showed significantly lower affective empathy compared to human participants. These findings highlight rapid advancements in language models' ability to simulate cognitive empathy, suggesting strong potential for providing effective virtual companionship and personalized emotional support. Additionally, their high cognitive yet lower affective empathy allows objective and consistent emotional support without running the risk of emotional fatigue or bias.