Most, if not all, firewalls are routers, but not all routers are firewalls, no.

These days, you usually see a few classes of devices in the commercial networking arena:

• Low-end routers, often with "odd" network interfaces, like a T1 interface, used as edge routers.
• Low-end switches that still generally have more features than you'd see in consumer switches, like 802.1Q VLANs and STP. You tend to find these in remote network closets attached back to a main switch with a fast trunk of some nature.
• Mid-range switch/router combination units. Sometimes you see these referred to as "Layer 3 Switches", which I think is a misnomer at best. These tend to have far more ports than a regular switch and also do routing. These are the real workhorses of a big network and are usually used as core routers.
• NAT devices/firewalls, which usually only have a handful of ports and often sit directly behind the edge routers. If the edge connection is ethernet, these can sometimes be used in place of an edge router.
• Access points. Many can actually support multiple SSIDs at the same time.

I'm sure I've left some stuff off.