You may consider the difference as the one between a dedicated server and the usual shared hosting. With VPS you receive all the power of dedicated hosting with the ability to have own set of services and customizable disk space. With shared hosting, you have limited administrative access and can't customize software configurations while with VPS you can perform any actions from your control panel and don't need to contact technical support with every single request.
Simply put, shared hosting
means your site will share the same server as many other sites. It's usually the cheapest option but comes with limited bandwidth, administration, and performance capabilities. VPS hosting
is a more premium option, with the ability for greater customization and increased performance.