I created an event

Clicked on "Manage Invitations"

Click on Invite a User

Search and find the user I want to invite

Click on the checkbox next to their name

Click on the Invite link

I'm returned to the Manage Invites page/module but the newly invited user isn't shown.

--

If I add the same user by going into their UserProfile and clicking on the [ADDTOEVENTS] button everything works fine.

Any idea on what I misconfigured?

Yes.  At the time of the original post, I had not yet installed the Group modules.  I installed the Group modules tonight however, and it still isn't working.

I did leave a couple of modules off like the birthday module.  It's installed to the system, but isn't installed to any pages.

Check that you are using it in Profile view and not list view (known issue) - that is one possible cause.