NEITHER!!!
Pick up a Preamplifier, a 5 Channel Amplifier, a 5.1 Dolby Digital Receiver, and some Acoustic Research HC1s. Computer gaming will never be the same, and you will be the envy of EVERYONE!
Okay, seriously though:
The Logitech Z560s sound very muddy. The subwoofer is extra boomy, so there will be times when you'll have to resort to using headphones. When you want to impress people though, this sub is hard to beat for under $300. All around they sound a bit muffled.
As for the Klipsch 4.1s. The sub has adequate power, but not as much as the Z560s. This is not the setup you want if you listen to a lot of hip hop (However, I prefer listening to my bookshelf system when I want to listen to CDs). The Klipsch sound great and low to mid range. They struggle a bit as the volume increases, and the pitch of the sound rises to a higher frequency. $300 is a bit much for getting speakers that have compromises at the extremes I enjoy most: hip hop bass, and music at high volumes.
If I were to choose though, Klipsch's overall clarity wins. The Z560s don't handle music very well, nor are the sounds produced clear in gaming applications. The Klipsch may not have as good of a subwoofer, but overall music quality is easier to enjoy, and it handles games well enough for your neighbors to complain.
But I have to stress...components are the way to go