The memory that holds the router's code is in flash memory. So when we flash a BIOS or a router we change the code to something else.
As to DD-WRT, for more there is no advantage. That is, without a background in networking folk can't use any feature that DD WRT might offer so it's as good as any old router.
What is the difference between a router and a "flash" router? I have a Netgear WNDR4500 router, but I am seeing sites that show this router as a Flash Router? Is this some sort of mod? I believe it is utilizing the Open Source firmware DD-WRT. If so, what are the advantages/disadvantages?