Next-generation (NextG) wireless networks are expected to require intelligent, scalable, and context-aware radio resource management (RRM) to support ultra-dense deployments, diverse service requirements, and dynamic network conditions. Digital twins (DTs) offer a powerful tool for network management by creating high-fidelity virtual replicas that model real-time network behavior, while large language models (LLMs) enhance decision-making through their advanced generalization and contextual reasoning capabilities. This article proposes LLM-driven DTs for network optimization (LLM-DTNet), a hierarchical framework that integrates multi-layer DT architectures with LLM-based orchestration to enable adaptive, real-time RRM in heterogeneous NextG networks. We present the fundamentals and design considerations of LLM-DTNet while discussing its effectiveness in proactive and situation-aware network management across terrestrial and non-terrestrial applications. Furthermore, we highlight key challenges, including scalable DT modeling, secure LLM-DT integration, energy-efficient implementations, and multimodal data processing, shaping future advancements in NextG intelligent wireless networks.