Can you ping when you connect on the non-browsing machine? this is a winsock problem. Use system file checker (Start, Run, SFC) to restore winsock.dll and wsock32.dll...hmm...can't remember if there are any others, but I don't think so. Just click on the option to
extract one file and enter the exact file name. If your system is setup with the cab files on the hard drive, point SFC to that directory when prompted. If the cabs are not on the disk, you will need the Win 98 CD to restore the files.
I had this exact problem on my sister's PC when i was doing something for her one day, and I ended up calling @home tech support, spoeaking with level one for awhile, then got bumped to level 2 to have this fixed. It was a quick 2 minute fix that took almost an hour when hold times and the 'dance' with tier 1 are factored in. I think you'll find that before you restore these files, you'll still be able to hit the mail POP servers, ping, tracert, etc. but nothing will come up in your broiwser window except an error about checking DNS settings. I would almost bet money (ALMOST, I said!) that you have corrupt winsock files.
Do a search on Google (ha ha...I'm a comedian...you can't browse!

) on your WORKING PC and you can probably find more info in case you need to restore more than just those 2 files.