A FirmwareEngineer behaves much like a SoftwareEngineer. The biggest difference is that firmware engineers work closer to the hardware on which their software is running. Because of this, a FirmwareEngineer tends to have more intimate knowledge of digital circuit design, things like hardware clock signals, timing diagrams, and other such things. In the wild, a firmware engineer can often be identified by the type of tools found in his office: an oscilloscope, a logic analyzer, a BDM or ICE, and probably too many cables for connecting all of the aforementioned tools.
What, no burn station?
Their vocabulary tends to be a little different, too. Including the ways in which they curse.
Bloody JTAG chain! What do you mean we've got f---ing 27ns latency from an interrupt!? And more colorful stuff.
Sprinkled among the aforementioned cables I have an assortment of hand-labeled EPROMs and Flash chips. And boards that have no cans. Cans? We don't need no stinking cans!
See also: EmbeddedSystemsEngineer, HardwareEngineer