Knuckles has it well-diagnosed.
Even if you installed OS 10.5.8, you would still struggle as 10.5 need a lot of RAM and you can't install more than1.25GB in that model PowerBook. Although Apple says you don't need that much, most of use around here feel that you need 1.5G or more for 10.5. Plus the video hardware in that PowerBook is realistically below what OS10.5 likes due to a lack of hardware support for one of Apple's key video technologies, CoreImage, in 10.5.
That PowerBook can still do some things well, such as routine office-type apps and email if (not web-based like gmail, hotmail, yahoo, etc.). The problem comes in web browsing where you are stuck with older, often non-supported versions of browsers and the inability to upgrade Flash beyond v10.2.102.64. And Knuckles gave you the lowdown on the iTunes issue if you must sync to an iDevice.
We just replaced a very capable G4 desktop for exactly those reason of web performance and the fact that our tax software no longer runs on a PowerPC Mac.
If you are in teh US, consider shopping for a refurb Mac here:
http://store.apple.com/us/browse/home/specialdeals/mac
We have six refurb Macs in the family plus a refurb iPad and two iPods. Everone one has been trouble-free, cosmetically unmarred, and they have the same one-year warranty as a new Mac and qualify for AppleCare extended coverage.
