Artificial intelligence (AI) has demonstrated significant potential in transforming healthcare through the analysis and modeling of electronic health records (EHRs). However, the inherent heterogeneity, temporal irregularity, and domain-specific nature of EHR data present unique challenges that differ fundamentally from those in vision and natural language tasks. This survey offers a comprehensive overview of recent advancements at the intersection of deep learning, large language models (LLMs), and EHR modeling. We introduce a unified taxonomy that spans five key design dimensions: data-centric approaches, neural architecture design, learning-focused strategies, multimodal learning, and LLM-based modeling systems. Within each dimension, we review representative methods addressing data quality enhancement, structural and temporal representation, self-supervised learning, and integration with clinical knowledge. We further highlight emerging trends such as foundation models, LLM-driven clinical agents, and EHR-to-text translation for downstream reasoning. Finally, we discuss open challenges in benchmarking, explainability, clinical alignment, and generalization across diverse clinical settings. This survey aims to provide a structured roadmap for advancing AI-driven EHR modeling and clinical decision support. For a comprehensive list of EHR-related methods, kindly refer to https://survey-on-tabular-data.github.io/.