Is SQL Server 2012 Management Studio "Express" any different? Are there any differences from the version of SSMS that comes as part of the SQL Server 2012 Enterprise Installer and the SQL Server 2012 Express Management Studio?
In the RTM release of SQL Server 2012 (and in previous versions), yes, they are quite different. The Express version was missing a bunch of functionality, most notably the ability to manage SQL Server Agent (even on non-Express instances). There are other things missing like Profiler but technically that is not part of Management Studio itself, just part of the suite of client tools.

Starting with SQL Server 2012 SP1, however, Management Studio Express now has full parity in terms of feature set. Essentially this means that full Management Studio functionality no longer requires a license. You can download the more recent version, SP2, here:

http://www.microsoft.com/en-us/download/details.aspx?id=43351

You want the file SQLManagementStudio_x64_ENU.exe or, if you're really old school, SQLManagementStudio_x86_ENU.exe.

As with previous versions, to get Management Studio Express, you'll need either Express with Tools, Express with Tools and Advanced Services, or stand-alone Management Studio Express.

I don't have a system handy without Management Studio proper to test this on, but they've blogged about this on the Internet (it was first announced with CTP4 back in September), so it must be true:

http://blogs.msdn.com/b/sqlreleaseservices/archive/2012/09/20/sql-server-2012-sp1-ctp4-now-available.aspx

I'm going to spin up a VM and install it and come back here and enumerate any differences I find. Unfortunately this is one of those cases where things have changed and pointing to a "duplicate" question doesn't help, since it deals with an older version...