Old CoyotePoint routers were just a commodity x86 motherboard with an ancient SSD instead of spinning rust. Junipers use a duo of x86 (routing engine) and ASIC (packet forwarding engine). Cisco has supposedly moved from that architecture to an ARM and ASIC pairing.

The ASIC is just a hardware offload for known routes. Unknown routes, admin work, and Ping packets are handled by the x86/ARM CPU. It's not too different from offloading graphics work to the ASIC on your graphics card, or your mining to your Bitcoin ASIC.