Large language models (LLMs) are increasingly used in decision-making tasks like résumé screening and content moderation, giving them the power to amplify or suppress certain perspectives. While previous research has identified disability-related biases in LLMs, little is known about how they conceptualize ableism or detect it in text. We evaluate the ability of four LLMs to identify nuanced ableism directed at autistic individuals. We examine the gap between their understanding of relevant terminology and their effectiveness in recognizing ableist content in context. Our results reveal that LLMs can identify autism-related language but often miss harmful or offensive connotations. Further, we conduct a qualitative comparison of human and LLM explanations. We find that LLMs tend to rely on surface-level keyword matching, leading to context misinterpretations, in contrast to human annotators who consider context, speaker identity, and potential impact. On the other hand, both LLMs and humans agree on the annotation scheme, suggesting that a binary classification is adequate for evaluating LLM performance, which is consistent with findings from prior studies involving human annotators.