Continue is an "open source autopilot in your IDE," designed to enhance coding with any LLM directly within the user's preferred development environment. It offers features like autocomplete, the ability to ask coding questions, natural language editing, and generating files from scratch. Continue is highly customizable and supports many LLMs and providers. It is "local-first," so users can bring their own local model to use even when their machine is offline. Continue also supports cloud-based LLMs, but users must bring their provide their own API key. Continue provides free access to a few cloud-based models but only on a trial basis.
TL;DR: Continue is a highly flexible, open source IDE tool for autocomplete and LLM chat. It's a strong option for users who want to bring their own models, particularly local models. Cody is a good option for users who want to use LLMs as a service, users who want free access to LLMs (via Cody Free), or for teams looking for solutions with enterprise security and support.