In this study, we propose the use of persistent homology -- specifically Betti curves for brain age prediction and for distinguishing between healthy and pathological aging. The proposed framework is applied to 100 structural MRI scans from the publicly available ADNI dataset. Our results indicate that Betti curve features, particularly those from dimension-1 (connected components) and dimension-2 (1D holes), effectively capture structural brain alterations associated with aging. Furthermore, clinical features are grouped into three categories based on their correlation, or lack thereof, with (i) predicted brain age and (ii) chronological age. The findings demonstrate that this approach successfully differentiates normal from pathological aging and provides a novel framework for understanding how structural brain changes relate to cognitive impairment. The proposed method serves as a foundation for developing potential biomarkers for early detection and monitoring of cognitive decline.