Run would be for the case where you have existing software on the Arduino (perhaps a "sketch") that is communicating with the simulink model; it could be the default interactive software running that allows you to query any arbitrary value on the Arduino.
Deploy To Hardware would be for the case where the Simulink model contains all of the information about how the Arduino has to behave and you wish to generate C or C++ code that is compiled to binary for installation on the Arduino, with the generated code being all that the Arduino is responsible for doing, with no interactive querying possible. This allows higher performance and also potentially allows the Arduino to run by itself without having to communicate with the host.
