Both transmit power from one rotating element to another and both can convert from one input RPM to a different output RPM. Now for the differences:
A chain drive uses a continuous loop of metal links, whereas a belt drive uses a loop of flexible rubber (frequently with strong tensile fibers or metal inside)
A chain drive requires a lubricating system. A belt drive does not.
A chain drive ensures synchronization between the input and output gears. Not all belts do; for example, a V-belt does not. There are toothed belts that do.
A belt drive is generally quieter than a chain drive
A belt drive has lower total installed cost than a chain drive