After a long fight with this silly program, I've won. Heres how:

Card Tested: ATI Radeon 9500 Pro
OS: WinXP SP1
Drivers: ATI Catalyst Release 4.6 (June 10, 2004)

Step 1: Patch the game
Go here.
Get the Riva TNT patch and apply it.

Step 2: Run FF7Config.exe
Click the 'Graphics' tab.
Make sure the NVIDIA box is checked.
Make sure that the 'TNT' circle is selected.
Select 'Direct3D Hardware Acceleration' in the 'Renderer' box.
Click OK.

Step 3 : Modify ATI 3D settings
Open the 'Display' control panel.
Click the 'Settings' tab.
Click the 'Advanced' button near the bottom of the tab.
Click the "ATI 3D" tab.
Move the 'Performance/Quality' slider to the 'Balanced' position.
Click 'Apply'.
Click 'OK'.
Close the 'Display' control panel.

Step 4 : Play the game

Notes:
FF7 hates FSAA. Period.
If you have any settings that force the application of FSAA upon the game, it will simply look like crap. The 'Balanced' setting under the 3D tab insures that FSAA is only turned on when an application requests it.
FSAA does not break the program, the game will run with FSAA on but some backgrounds will appear maligned. Oddly enough, the combat portion of the game is not hurt by FSAA, quite the contrary, it improves the display of text as well as other textures.

On a side note, for a patch to change the MIDI music to that of the actual FF7 playstation game, go here.

Thats all folks.