This is one computer? And you can get out on the internet when you boot into Win2000, but not when you boot into Win98?
Look at your Win98 modem settings. Make sure they are correct in device manager. Make sure there are no IRQ conflicts. Then look at your browser settings and see if it's still trying to connect via modem instead of your network card. If you are using DSL you may need to install the PPoE software to run DSL in Win98. If you are using cable, then make sure those settings in Win98 are correct.
Hardware should be controlled by software. The boot process determines which OS you are using ; I think a second nic would only create more problems.
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Just as a side, I am a firm believer in hdd drawers, where you purchase ide drawers and mount the shell to your computer, then place the drive in the pull out drawer so that you basically insert the OS you wich to use with the computer and then power up the computer. You're always good-to-go and do not experience any of these OS confusions in dual-boot computers.