Analogical reasoning is considered core to human learning and cognition. Recent studies have compared the analogical reasoning abilities of human subjects and Large Language Models (LLMs) on abstract symbol manipulation tasks, such as letter string analogies. However, these studies largely neglect analogical reasoning over semantically meaningful symbols, such as natural language words. This ability to draw analogies that link language to non-linguistic domains, which we term semantic structure-mapping, is thought to play a crucial role in language acquisition and broader cognitive development. We test human subjects and LLMs on analogical reasoning tasks that require the transfer of semantic structure and content from one domain to another. Advanced LLMs match human performance across many task variations. However, humans and LLMs respond differently to certain task variations and semantic distractors. Overall, our data suggest that LLMs are approaching human-level performance on these important cognitive tasks, but are not yet entirely human like.