Your ISP normally only allows you ONE public IP w/ your home service. In order to share that one public IP w/ multiple computers, you MUST have a router! A router maps that one public IP to many LOCAL IPs on your own home network. What's happening is that each time one computer gets connected, it grabs the only IP available, the public IP. That locks the other computer(s) out.
The reason your WAP works at work is either because it's actually using a router somewhere else in the building you don’t know about or see, OR, the service at work supports MULTIPLE public IPs.