Why is my @@Version unchanged after upgrading?
I found the problem: The installer I had was just for SQL Server Management Studio, as @MarkSinkinson pointed out.

If you have this same problem, verify you have the correct installation file (not SSMS only, or an incompatible Edition).

Other helpful things to check, from comments originally left on the question:


You may have multiple instances installed, upgraded one of them, and connected to another. Or the upgrade didn't take (this can be easy to dismiss without noticing in the final step of the upgrade wizard). Please note that 2008 and 2008 R2 are completely separate, independent releases.
Make sure that you are connected to right instance in any case.
Check you didn't uncheck the database engine installation option