It is good you mentioned a google.com ping works. This helps narrow the problem down.
As far as I'm aware pings don't use ports.
The internet protocol HTTP uses port 80.
So, the thing that comes to mind is port 80 is getting blocked on your machine, and the "portless" ping traffic makes it through fine.
Try searching for google by specifying the port number. type www.google.com:80 into your URL bar of your browser. The :80 is specified by default but it's worth explicitly typing it for testing purposes. If it doesn't work then port 80 is either closed or something down the line is preventing the connection.
Also, with the ethernet cable plugged in, try typing netstat -an into cmd and see if there are any foreign port 80's, if so then port 80 is indeed open. If not then port 80 is probably closed.
If port 80 is blocked it is more than likely on your router since you don't have a firewall installed. Typically port rules are assigned to IP addresses so try giving your XP machine another IP address within your network range.
I would also try updating the network interface card drivers too if you haven't already.
There's many of deeper things it could be but try those first.