A game engine and a graphics engine, while often used interchangeably, serve two different purposes in the field of game development.
A game engine is a software framework designed to facilitate the creation of video games. It provides a suite of systems working together to help developers create games more efficiently. These systems typically include rendering for graphics and animations (the graphics engine's role), physics engines, sound, scripting, animation, artificial intelligence, networking, and more. Examples of game engines include Unity, Unreal Engine, Godot, etc.
On the other hand, a graphics engine, also known as a rendering engine, is a component of a game engine that is responsible for rendering 2D or 3D graphics. A graphics engine takes graphical assets (like models, textures, lighting information) and outputs them to your screen in real-time. It doesn't handle things like game logic, AI, or physics - it's purely concerned with drawing visuals.
Here is a simple analogy: If you are building a car, the game engine is like the entire car - it has an engine, wheels, seats, etc., making it fully functional. The graphics engine is like the car's engine. It plays an important part but isn't a car on its own.