[Not really an answer as I'm just stating the obvious but I don't have enough rep to comment.]
The four driver files are the ODBC driver, the JDBC driver, and the 32 and 64-bit OLEDB drivers. The ODBC driver is of course the correct one but there's a small check box in one of the installer dialogs which allows you to choose 32-bit and/or 64-bit. Did you select the correct one?
I've just tried this on Windows Server 2012 and Windows 7 and both work exactly the same for me without any extra steps. The driver I have is called
C:\Windows\System32 (and I also have a 32-bit version in
C:\Windows\SysWOW64). I can connect from both the 32-bit and 64-bit ODBC Data Source Administrator tools although I got my password wrong several times so that's certainly worth checking.
Have you also checked that the advanced options are the same, SSL for instance?
Possibly there's a security setting on your Server box that's preventing the connection but on 2012 Server there didn't appear to be anything by default.