Does your router have 1000mbit ports(gigabit router) or just 100mbit ports??
Only thing I can think of off hand is that maybe your router is a gigabit router and your modem only has a 100mbit port, so when you connect to the router your NIC is trying to negotiate 1000mbit link but when you connect to the modem directly it is linking up at 100mbit, and perhaps your NIC or drivers for the NIC are faulty when trying to link at 1000mbit speed.
These speeds don't have anything to do with the bandwidth you get from your ISP, they are the theoretical maximum throughput speeds of a network interface.
Quickest way to check is to open task manager and open the Networking tab, it should list all network interfaces and what link speed they are connected at, ie. 100mbit or 1000mbit, see what it says when connected to router versus what it says when connected to the modem directly.
However, regardless if that is the problem or not, it sounds like your NIC could be bad, I see them go bad all the time, but usually they just perform bad and die, they don't reboot the PC.
Add-in NIC's don't cost much, they are easy to install into a PCI slot, and most of the time Windows has drivers for them so you don't have to install drivers from a disk.
If none of that helps you might check with AceTW(formerly Malfunction{BAM}) at
http://gtr.smfforfree.com/ That is the new location for his Gamers Technical Resources forums.