This is a great question. From my experience, a styleguide is merely reference material or parameters to follow when designing. Most styleguides include color, typography, and elevation information.

With that said, a design system takes a lot of the same things included in a styleguide but goes further by adding the code and documentation elements. Mostly the code element which allows design tokens to be created and used (ie color nicknames). These nicknames not only allow a shared language between design and engineering but they create an easy way to maintain or update them later, just by changing the value of the nickname.

A design system is a holistic solution to design across multiple teams and features within a product that includes information you can find in a styleguide but partners that information with a code element and contextual guidance.