I think the differences can be boiled down into a few key points: First, boards that are designed for ASIC emulation can have several, very large FPGAs that usually provide mostly pure fabric logic resources (as opposed to DSP slices, hard IP cores, and transceivers) with lots of interconnections between them vs. "normal" dev boards which usually have just one mid-range FPGA. Second, boards intended for ASIC emulation usually don't have much in the way of peripherals, usually just broken out IO pins and serializers and possibly some memory modules, unlike normal FPGA dev boards that usually provide a 'kitchen sink' of various peripherals. ASIC emulation boards will also contain dedicated nets for clocks, resets, and other global signals. The manufacturers of ASIC emulation boards may also provide various software packages and HDL IP that can aid in utilizing an ASIC emulator.
If your design fits on a single FPGA, then it doesn't make much difference what board you get so long as the FPGA is big enough and it has whatever connectivity you need. However, if it has to be segmented across several FPGAs, then you will probably want to get a board specifically intended for ASIC emulation.