To be clear, FSOD and "PSOBB.exe has stopped working" are the same thing. FSOD happens because the DC and GC don't have an underlying OS; the game is in full control of the hardware, so when it crashes, there's no "desktop" or anything to crash to, so the game just freezes. PC PSO versions run within an OS, so when the game crashes, the program exits, but the rest of the programs on your PC continue to work.
Most of the trolls who went around FSODing people have left PSO to infest newer games, but the game still likes to shit itself regularly, so yes, you're still going to crash sometimes, because Sega sucks at programming.