<p>For quite a while, I thought that Free Software was Open Source Software.  I've found out that this view is incorrect, and that Open Source Software is not necessarily Free Software.  I honestly can't see any differences.</p>

<p>What am I missing here?  What are the distinguishing traits of both parties?</p>

Both are basically the same, except the free software movement puts more emphasis on the freedom to modify and redistribute the code. For example, GNU GPL would be more "free" than MIT licence, because MIT license does not enforce copyleft and thus someone can develop closed-source software based on the code.
See also
Wikipedia chapter about this
, which mentions Microsoft shared source inititive, that can provide you with very unfree source code of their applications.