Search-based reasoning and chain-of-thought (CoT) prompting offer two contrasting inference-time paradigms for language models: CoT commits to a single linear trajectory, while search explores alternative partial reasoning paths before selecting an answer. In this work, we study this trade-off using A* as a controlled instantiation of heuristic-guided search. Across StrategyQA, LogicQA, HotpotQA, GSM8K, and MATH, and across three model scales, we find that search is not a universal upgrade over CoT; instead, the two strategies solve substantially different subsets of instances. To explain this complementarity, we introduce a Reasoning Topology Framework based on structural features computable before generation. A derived branching coefficient is strongly associated with when search helps across mixed-benchmark topology clusters, suggesting that the CoT–search boundary is partly explained by problem structure. We further train a topology-aware router that selects between CoT and A* without observing generated traces. Despite this restriction, it outperforms post-hoc routers based on semantic entropy and LLM-as-critic judgments. We attribute this to a Fluency–Correctness Asymmetry: fluent traces are often incorrect, making trace-level adjudication unreliable. Overall, our results suggest that adaptive reasoning should route by problem structure, deploying search only when the input topology warrants its higher cost. Code: https://github.com/newcodevelop/astar-cot-reasoning.