Run the installer, click "Installed Products...". This will give you a more detailed list of all installed components of the client install, e.g., drivers, SQL*Plus, etc.
Typical Oracle installations will store inventory information in C:\Program Files\Oracle\Inventory, but figuring out the installed versions isn't simply a matter of opening a text file.
This is AFAIK authoritative, and shows any patches that might have been applied as well (which running the utilities does not do).
EDIT:
A CLI option would be to use the OPatch utility:
c:\> path=%path%;<path to OPatch directory in client home, e.g., C:\oracle\product\10.2.0\client_1\OPatch>
c:\>set ORACLE_HOME=<oracle home directory of client, e.g., C:\Oracle\product\10.2.0\client_1>
c:\>opatch lsinventory
This gives you the overall version of the client installed.