A server-based load balancer runs on a standard operating system on a PC box.
Special software runs on top of the operating system to perform the load-balancing functions.
ASICs are chips with the software for a specific function hard-wired into the circuitry, rather than a chip with a general instruction set, such as a Pentium. The advantage of having the software hard-coded into a chip is speed. Without the abstraction layer of software on top of the operating system on top of machine code on top of a general processor, you can perform a task much more efficiently. The drawback is a lack of flexibility, because an ASIC chip cannot be reprogrammed. If the history of Layer 3 routers is any indication, hardware-based products will probably prevail in the long run.