The functional connectivity network as a graph-theoretical model has been extensively applied in brain network analysis, where brain regions are represented as nodes and functional connections between these regions are considered as edges. Neuroscience research indicates that cognitive processes in the brain are accomplished not by isolated brain regions, but through subnetworks composed of diverse brain regions. However, conventional methods based on brain functional connectivity typically focus on the local node characteristics of individual brain region, while neglecting the global network properties which also reflect the symptoms of cognitive disorders. To overcome this limitation, we propose a motif-aware brain functional connectivity network analysis method (MA-BCN) that captures high-level information beyond individual brain regions for cognitive disorder diagnosis. Specifically, we introduce a novel algorithm for detecting motifs within brain functional connectivity networks. Subsequently, we construct a Motif Graph (MG) to effectively delineate the relationships between these motifs. We also propose an enhanced node representation learning strategy that integrates information from brain region nodes, motif features, and prior knowledge to capture complex correlations at multiple levels within brain networks. The embeddings generated from the comprehensive brain network are utilized to diagnose cognitive disorders. Experiments on multiple diagnostic tasks for brain cognitive disorders demonstrate that the proposed MA-BCN method achieves significant performance gains, reaching up to 91.90% accuracy and 93.23% AUC in distinguishing NC from AD, and outperforming existing state-of-the-art approaches across all tasks.