No internet when connected to the router--internet when there is a direct connection the original computer.
Who's your Internet service provider? Many cable ISP's, like Comcast, only allow one device connected to the cable modem at a time. If you switch the device, like you did when you swapped your computer for the router, the cable modem will not talk to the new device until you power cycle the modem. Have you done this?
The reason you have to do this is the cable company downloads a config file into the modem when it initializes, and part of that config says how big the internal table of attached devices can be. It's usually one unless you pay for extra IP addresses. Once you fill up that this so-called CPE table with your computer's MAC address, the only way to add a new device (like the router) is to wipe the table first with a power cycle or reset. If you're using one of those telephony cable modems for use with digital phone service, use the reset switch in the back of the modem, simply removing the power won't work as these modems have internal batteries.