You are still connecting a computer to one of LAN ports of W8980? What's the IP address obtained? 169.254...? Can you still detect Wi-Fi for W8980? If to reset it about 15 seconds makes no...

Probably too many clients access internet at the same time, which causes such a problem... What's the maximum number of clients supported by your main router? RE200 is running in latest firmware?

The main router is secured with WPA/WPA2 or WPA/WPA2-personal? You have to verify if the router can be secured with WPA/WPA2 and if you can connect to it successfully...

There is a possible solution here. Update chipmaker's driver. http://forum.tp-link.com/showthread.php?79513-Network-crash-when-heavy-downloading-unable-to-reboot-network-or-shut-down-computer

How about status of ADSL light and Internet light when connection drops? Can you login its web interface at that time? Firmware of your W8968 is the latest version? Try to disable UPNP.

Since the modem can be identified but no internet, did you try to manually configure APN, dial number etc.?