The main difference is sound. An open dump pipe (also called a "screamer pipe") will be very loud when you reach your target boost since you will basically have a big exhaust "leak" that bypasses the rest of your exhaust system. So when you hit full boost you will hear your car get significantly louder while a recirculated wastegate is pretty much inaudible. You can hear what this sounds like on a wrx in this video:
Fast forward to ~3:40.