Straight from Wikipedia:
Routers connect two or more logical subnets, which do not necessarily map one-to-one to the physical interfaces of the router. The term "layer 3 switch" often is used interchangeably with router, but switch is a general term without a rigorous technical definition. In marketing usage, it is generally optimized for Ethernet LAN interfaces and may not have other physical interface types. In comparison, a network hub does not do any routing, instead every packet it receives on one network line gets forwarded to all the other network lines.
Modem (from modulator-demodulator) is a device that modulates an analog carrier signal to encode digital information, and also demodulates such a carrier signal to decode the transmitted information. The goal is to produce a signal that can be transmitted easily and decoded to reproduce the original digital data. Modems can be used over any means of transmitting analog signals, from driven diodes to radio.
The most familiar example is a voiceband modem that turns the digital 1s and 0s of a personal computer into sounds that can be transmitted over the telephone lines of Plain Old Telephone Systems (POTS), and once received on the other side, converts those 1s and 0s back into a form used by a USB, Ethernet, serial, or network connection. Modems are generally classified by the amount of data they can send in a given time, normally measured in bits per second, or "bps". They can also be classified by Baud, the number of times the modem changes its signal state per second.
So the difference being a router connects two or more subnets in a network, most likely via LAN while a modem modulates and demodulates a carrier signal such as a phone line.