Every router has a unique MAC address and I think its identity is saved in your network settings when you power down the computer on your home network.
When you power up at work, the computer reads those settings, but detects a different MAC ID on the wokplace router. The 'limited connectivity' thing may be a security precaution built-in to the networking software. Rebooting the router sorts things out in the same way as would happen if you changed router hardware at home.
In a similar way, if you added a new PC at home, you're expected to power up everything (including printers, scanners) and run the home networking wizard again.
As a separate issue, your workplace router can be set up to lock out 'drive-by Wi-Fi surfers', who attempt to do exactly what you're doing with your laptop, by means of a list of "authorised devices". There should be a page in the router settings menu for this. If not auto-detected, you will need to find the MAC addresses of your laptop and mobile phone to input them manually.