The role of aerodynamics is primarily three fold:
1. Low drag for higer speeds with less power
2. High down force for ability to corner at high speeds, also for general stability.
3. Cooling (engine, brakes, transmission, electronics, etc)
The reason to have the bottom of the car covered is indeed to promote high speed laminar flow under the vehicle. The adds down force. Running a flat smooth underbody with no rear diffuser leads to increased drag when the underbody flow mixes and forms turbulence behind the car. A rear diffuser mitigate this by mixing the underbody flow into the vehicles wake.
OK...back the the M3...Given that the car does have a fairly smoothed flat bottom there is a definite need for a diffuser. Although the design on the exterior of the car looks nowhere near as dramatic as on many other cars (F430 for example) I suspect it does have significant function. I also remember reading (I think from the long BMW press release) that the entire design of the underbody and diffuser also contributes to cooling its trick M differential. Although not based on facts, published statements, CFD or wind tunnel evidence BMW M is well known for designs that emphasize function over form and I seriously doubt they would have a non-functional rear diffuser on the car.