You add any computer to a home network...by running the XP Network Wizard, ensuring that all computers belong to the same workgroup (MS Home is the default and works, but any name would do).
AFAIK, SP2 and SP3 should not affect your home network. I have two systems and ran one on Computer A and the other on Computer B for a time...no problems noted.
The home network is the key...and each computer must reflect the exact same protocols in order for the network to function effectively. So, that's what I would check first.
If neither system properly reflects a network connection and that connection doesn't reflect the same protocols as the other...I would change that. That may mean using the Network Wizard again to set up the connection (which takes all of about 2 minutes). When asked for type of connection..."residential gateway" is what a router is.
Of course, it won't hurt to check Device Manager for drivers for the network devices, just to cover all basic bases.
My connections reflect the following when I click on Properties: Client for MS Networks, File & Printer Sharing for MS Networks, QoS Packet Scheduler, my firewall, NWLink NetBIOS, NWLink IPX, etc...and TCP/IP. You may not necessarily reflect all of those...just try to ensure that what is reflected for the system that works/connects...is what you set up in the system that doesn't.
My two NWLink entries are probably not needed by you. I found out, by trial-and-error, that I need those to facilitate communication between my systems, although I had previously been told that I didn't reuire these for my net to work properly

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Louis