Are the websites that cause the music to skip Flash-heavy? Or is the skipping behavior consistent across all sorts and forms of websites? Have you tried using a different browser?
It might be something related to the Flash plugin, or possibly hardware-accelerated Flash. Could you please post what version of ATI drivers and Flash you've got installed? I'd also recommend checking the SMART data for your HDD for anything fishy, like any unusually high values. I used to run an PII X3 720 until mid 2011 (though quad unlocked and mildly OC'd), and those things should easily handle browsing and music at the same time, even when using fancy input and output plugins in Winamp (in_mpg123/in_mad, out_asio). I used to rip music on the same machine using EAC and LAME while browsing and I don't remember those causing music drop-outs.
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