What are the specific differences between .msi and setup.exe file?

Question

An MSI is a Windows Installer database. Windows Installer (a service installed with Windows) uses this to install software on your system (i.e. copy files, set registry values, etc...).

A setup.exe may either be a bootstrapper or a non-msi installer. A non-msi installer will extract the installation resources from itself and manage their installation directly. A bootstrapper will contain an MSI instead of individual files. In this case, the setup.exe will call Windows Installer to install the MSI.

Some reasons you might want to use a setup.exe:

• Windows Installer only allows one MSI to be installing at a time. This means that it is difficult to have an MSI install other MSIs (e.g. dependencies like the .NET framework or C++ runtime). Since a setup.exe is not an MSI, it can be used to install several MSIs in sequence.
• You might want more precise control over how the installation is managed. An MSI has very specific rules about how it manages the installations, including installing, upgrading, and uninstalling. A setup.exe gives complete control over the software configuration process. This should only be done if you really need the extra control since it is a lot of work, and it can be tricky to get it right.

.msi files are windows installer files without the windows installer runtime, setup.exe can be any executable programm (probably one that installs stuff on your computer)

MSI is an installer file which installs your program on the executing system.

Setup.exe is an application (executable file) which has msi file(s) as its one of the resources. Executing Setup.exe will in turn execute msi (the installer) which writes your application to the system.

Edit (as suggested in comment): Setup executable files don't necessarily have an MSI resource internally