Router describes the general technical function (layer-3 forwarding) or a hardware device intended for that purpose, while gateway describes the function for the local segment (providing connectivity to elsewhere). You could also state that "you set up a router as gateway". Another term is hop which describes the forwarding in between subnets.
The term default gateway is used to mean the router on your LAN which has the responsibility of being the first point of contact for traffic to computers outside the LAN.
It's just a matter of perspective, the device is the same.