Neurological disorders pose major global health challenges, driving advances in brain signal analysis. Scalp electroencephalography (EEG) and intracranial EEG (iEEG) are widely used for diagnosis and monitoring. However, dataset heterogeneity and task variations hinder the development of robust deep learning solutions. This review systematically examines recent advances in deep learning approaches for EEG/iEEG-based neurological diagnostics, focusing on applications across 7 neurological conditions using 46 datasets. For each condition, we review representative methods and their quantitative results, integrating performance comparisons with analyses of data usage, model design, and task-specific adaptations, while highlighting the role of pre-trained multi-task models in achieving scalable, generalizable solutions. Finally, we propose a standardized benchmark to evaluate models across diverse datasets and improve reproducibility, emphasizing how recent innovations are transforming neurological diagnostics toward intelligent, adaptable healthcare systems.