If you are not playing games and don't have problems with the X1600 then you have no need to upgrade. Of course need and want are different, so I looked at your computer specs and at the video card specs and the 8600GT version does not require extra power while the 8600GTS does. The GTS requires at least a 350 watt power supply with 18 amps on the 12 volt rail. You do have a 350 watt power supply, but I don't know how many amps are on the 12 volt rail.
Whatever you do, do not purchase a version that has DDR2 memory. they are slower than the versions with DDR3.