ieee,
When I hear the expression "VLSI Engineer", I think of someone who is involved with the actual process of producing an ASIC. That can be someone involved in the physical fabrication process including mask making and the various steps involved in creating the physical transistors and metal layers on a wafer.
By "ASIC Engineer" I think of someone who is involved in the creation and verification of the functional aspect of the chip.
According to my definition, I believe that ASIC Engineers on the whole make more money than VLSI engineers. However, it seems like both jobs are on the decline in the US. Many of these types of jobs are being outsourced. A lot of ASIC designers in the US are transitioning to FPGA or high-level system design.
Just my 2 cents...
Radix