Humans organize knowledge into compact conceptual categories that balance compression with semantic richness. Large Language Models (LLMs) exhibit impressive linguistic abilities, but whether they navigate this same compression-meaning trade-off remains unclear. We apply an Information Bottleneck framework to compare human conceptual structure with embeddings from 40+ LLMs using classic categorization benchmarks. We find that LLMs broadly align with human category boundaries, yet fall short on fine-grained semantic distinctions. Unlike humans, who maintain ``inefficient'' representations that preserve contextual nuance, LLMs aggressively compress, achieving more optimal information-theoretic compression at the cost of semantic richness. Surprisingly, encoder models outperform much larger decoder models in human alignment, suggesting that understanding and generation rely on distinct representational mechanisms. Training-dynamics analysis reveals a two-phase trajectory: rapid initial concept formation followed by architectural reorganization, during which semantic processing migrates from deep to mid-network layers as the model discovers increasingly efficient, sparser encodings. These divergent strategies, where LLMs optimize for compression and humans for adaptive utility, reveal fundamental differences between artificial and natural intelligence. This highlights the need for models that preserve the conceptual ``inefficiencies'' essential for human-like understanding.