Learned database components, which deeply integrate machine learning into their design, have been extensively studied in recent years. Given the dynamism of databases, where data and workloads continuously drift, it is crucial for learned database components to remain effective and efficient in the face of data and workload drift. Adaptability, therefore, is a key factor in assessing their practical applicability. However, existing benchmarks for learned database components either overlook or oversimplify the treatment of data and workload drift, failing to evaluate learned database components across a broad range of drift scenarios. This paper presents NeurBench, a new benchmark suite that applies measurable and controllable data and workload drift to enable systematic performance evaluations of learned database components. We quantify diverse types of drift by introducing a key concept called the drift factor. Building on this formulation, we propose a drift-aware data and workload generation framework that effectively simulates real-world drift while preserving inherent correlations. We employ NeurBench to evaluate state-of-the-art learned query optimizers, learned indexes, and learned concurrency control within a consistent experimental process, providing insights into their performance under diverse data and workload drift scenarios.