Have you actually "Connected" to the wireless network? In other words, have you run the wireless adapter's software, scanned for "Available Wireless Networks", found and highlighted the one that matches your router's SSID (network name) and selected the option to "Connect" to it? Once "connected", the wireless adapter should get its IP address by DHCP from the router and you should have network and internet access.
There is a difference between the Wireless Network Adapter and Router communicating with a good signal and "Connecting" (as in "logging in") to your Wireless network.
The "Limited or No Connectivity" indication means that the adapter has been unable to get its IP address from the router and Windows has allocated a default "169.254" address. This won't let you connect to the router or get online.