Do you have a broadband router or do you plug in directly to the modem? Your computer most likely uses DHCP to get its IP address from the router and then the router gets it IP information from the modem. If you are not using a router then the computer gets its IP information directly from the modem.
Your computer may be having a problem getting its IP information from the router or modem. If you are using a router try going directly into the modem and bypass the router. Then see if you have the same problem after a couple of reboots. If so then your router may have a problem giving your computer the IP information it needs to connect to the internet.
If you are not using a router then your modem may not be giving the computer the proper information. When it does not connect you should check the IP information and make sure you are getting an IP address.
If the IP address begins with 169.254.x.x then that is the Windows autoconfig IP address that is uses when it can’t get one from a DHCP device. If the IP address is all zeros or uses the autoconfig IP then you can try to renew your IP address to see if that makes a difference.
There is also a chance your computer could be having issues obtaining an IP address but I would check the router and modem first.
If the port on the cable modem and the Ethernet cable works with your other computer then you know its not the port or the cable. The only thing left is either the port is bad hardware wise or it is not enabled or recognized within Windows. The first thing to do is to open your network connections and make sure your local area connection is enabled. You can right click on it to check. You can also check in Device Manager to make sure your Ethernet network card is installed and enabled. You may want to update the driver if it is enabled and shows that its working properly.
If your Ethernet connection on the laptop has a link light make sure it is lit up and also lit on the port on the other end. It is usually green for a good link and amber for no link. Sometimes the port will only have one light and it may be amber for a good connection so be aware of that. You can also try plugging the laptop into another device like a router or hub and check for link lights to get a second opinion on the port. If it turns out that it is bad you can install an Ethernet PC card into an open slot and use it that way.