... and the other is a smaller connection lined with white inside
That's a USB port. Some modems use USB instead of the traditional Ethernet ports; the expectation being (I assume) that the modem will be connected directly to a computer (no router in the picture) through USB.
Unfortunately though, most routers use an Ethernet port for their modem-side connection. Your best bet is to see if your friend can exchange the USB modem for an Ethernet modem, as a hard-wired Ethernet connection between the modem and router usually yields more reliable connectivity.