Physical Servers, also known as Dedicated Servers, are standalone servers where the server resources are not shared with other users on the server or on a network of servers.
Mini Servers (VPS) are physcial servers that are split into containers which allow users to install different operating systems and software etc. in a container. Each container is private so other users do not have access to other containers but users do share resources such as RAM, Bandwidth, Disk Space.
In terms of resources and cost Mini Servers sit in the middle between Shared Web Hosting and Physical/Dedicated servers. With Mini Server VPS the cost of the physical server is being shared by the users on the server. Each user can install their preferred operating system and software to run Websites much the same as a physical/dedicated machine but without the higher cost.
Using a Mini Server VPS instead of a physical/dedicated machine where possible will reduce costs. However the decision to use a Mini Server VPS instead of a physical/dedicated server will depend on the resources required to run the Website or application and whether the additional cost is warranted or not.