<p>We are working on a winforms app in Visual Studio 2005 and the setup project we created output both an MSI and an EXE.  We aren't sure what the EXE file is used for because we are able to install without the EXE.</p>

It's a bootstrapper that checks to make sure that the .NET Framework is installed, before launching the MSI. It's pretty handy.
I suggest using something like SFX Compiler to package the two together into one self-extracting .exe and then launch the extracted setup.exe. This way you retain the benefits of the bootstrapper, but your users only download a single thing.
Edit
: also see
The official line:
MSDN documentation
Some bootstrapper customization:
some guy's blog post about what he did