What are the advantages to a separate processor when there is an FPGA present?
Generally speaking, a hard core will run faster, use less power, and will use less chip area. 

That may mean that it's cheaper or better to use a smaller FPGA and a hard processor (either on one chip or on two). 

Some FPGAs have hard cores on chip (for example, the Xilinx Zynq series), so you can use one or both of the two hard ARM9 cores, and/or you could implement a PicoBlaze or MicroBlaze soft (or some other type) processor.  

There are also most likely business differences associated with licensing IP for a soft core compatible with (say) an ARM toolchain.