The big advantage of an afterburner is that you can significantly increase the thrust of the engine without adding much weight or complexity to the engine. An afterburner is nothing but a set of fuel injectors, a tube and flame holder that the fuel burns in, and an adjustable nozzle. A jet engine with an afterburner needs an adjustable nozzle so that it can work both with the afterburners on and off.
The disadvantage of an afterburner is that it uses a lot of fuel for the power it generates. Therefore most planes use afterburners sparingly. For example, a military jet would use its afterburners when taking off from the short runway on an aircraft carrier, or during a high-speed maneuver in a dogfight.
Well I looked this up, and can conclude some main things:
- What you said about weight seems rather trivial.
- One problem we can wonder about is how much O² is left in the exhaust? It may be less than in a jet (fighter) engine.
- However I don't really know how te explain this, all afterburners apparently need a veriable width of exhaust at the ends. An afterburner heats exhaust air even further, therefore increasing thrust. I think thereby that veriable exhausts are forbidden in F1
- Afterburners are apparently designed to give thrust by pushing gasses out. That is actually not really what F1 needs. We need rotation power for the wheels. Thrust would be however quite interesting, as the car is pushed forward without demanding acceleration grip from the tyres.
BUT... since turbines (and therefore I think also thrustpower) is forbidden
would be nice though, seeing the back of the car glowing red when accellerating from slow corners