Challenges and Applications of Large Language Models: A Comparison of GPT and DeepSeek family of models

Large Language Models (LLMs) are transforming AI across industries, but their development and deployment remain complex. This survey reviews 16 key challenges in building and using LLMs and examines how these challenges are addressed by two state-of-the-art models with unique approaches: OpenAI's closed source GPT-4o (May 2024 update) and DeepSeek-V3-0324 (March 2025), a large open source Mixture-of-Experts model. Through this comparison, we showcase the trade-offs between closed source models (robust safety, fine-tuned reliability) and open source models (efficiency, adaptability). We also explore LLM applications across different domains (from chatbots and coding tools to healthcare and education), highlighting which model attributes are best suited for each use case. This article aims to guide AI researchers, developers, and decision-makers in understanding current LLM capabilities, limitations, and best practices.