Ok. Here are the actual facts: Yes, you can connect the modem to the wii via usb 'electrically'. The cables are available, they will connect. Nothing more than a simple usb a male to usb b male cable which is used by a wide number of devices out there.
The problem is that the usb interface the modem uses is not compatible with the limited 'device drivers' that the Wii contains to utilize usb ethernet adapters. And before anyone asks, it is WAY too difficult to add in the required drivers for a wide number of reasons to the point it is much cheaper and simpler to either get a wireless router or a compatible usb ethernet adapter.
To give a valid analog, it's the same as if you tried to hook a usb device up to a windows machine but didn't have the drivers for it and there is no given way, shape, or form to get working drivers on the system. It'll connect, yes. It'll even see a connected usb device. But that is as far as it'll go. No further.
As for the usb ethernet adapter problem given by the OP: If you are using it on the same modem and switched the connection from your pc to the Wii, make absolute certain that you unplug the modem, wait about a minute, then plug it back in. Should fix it.