Just as sidenote: There are also mixed systems. Those how have a cpu and an FPGA/CPLD. Yes, there are cpu cores for FPGA. Ever tried those tools ? the compiler, debugger ? I don't intend to put them down. I never tried yet. I'd take me at least a month to have a closer look.
As to the advantage of a mixed setup : You have the cpu and the tools you're familiar with plus you get a configurable digital hardware that can do timing stuff, that the cpu cannot.
FPGA's are useful for logic that is too proprietory/complex to use COTS parts, or high-speed logic that just won't cut it in software. Or perhaps there are several functions that would require a selection of COTS parts but would fit nicely into a single FPGA. Or perhaps real-estate is critical.
A few random thoughts...
If you're looking to glue a few small peripherals onto a small micro, then I'd suggest you go down the uC/uP path. It'll probably be cheaper and require much less development effort.
If you have some (more complex or high speed) logic that could benefit from a bit of software (to reduce development complexity) then perhaps an FPGA with embedded soft uC is the go.
Is there scope for the design to evolve - will an FPGA provide a platform that allows firmware-only upgrades to functionality?
Then you need to look at manufacturing - how does the FPGA compare with uC/uP solutions in terms of PCB requirements (layout, layers) and SMD technologies (eg. BGA)? Should it be servicable in the field? What's the lifetime of this product (can you still get the same FPGA in 10 years?). How much will the unit cost to make?
And there's always - want to learn something new and have some fun? ;)