In my mind, there are two major reasons for hybridizing a car, discounting the plug-in kind:
1. You can seriously downsize the engine, while still keeping enough power to pass and accelerate reasonably.
Running at high load (low vacuum) is most efficient. With a big engine, to maintain speed and not accelerate it's necessary to keep the throttle plate almost closed. With a hybrid, the engine is small enough that you can nearly run WoT on a level road and maintain constant speed. When passing, the electric motor comes on and gives the acceleration. Additionally, Toyota's Prius engine uses something similar to the Atkinson cycle, which gives it nearly the same volumetric efficiency as some of the best diesels on the road, with the downside of losing a lot of low-end torque, which is made up for by electric assist at low RPM.
Insight G1 - 1.0 liter
Insight G2 / Civic Hybrid - 1.3 liter
Prius - 1.5 liter / 1.8 liter with Atkinson cycle
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2. It allows for the recovery and prevention of lost energy.
When slowing down, the hybrid motor brakes the car and fills up the batteries, storing energy that would otherwise have been used to convert brake pads into brake dust.
When stopped, a hybrid shuts off its engine. Although it's very possible to do "auto-stop" with a conventional car (and some do now, e.g. F150 ecoboost) traditional starter motors are not designed for that kind of duty cycle and must be upsized. With a better starter, you may as well provide assist with it unless it's on a particularly large engine.
Hybrid motors are more efficient at generating electricity than traditional alternators, and eliminate the need for this accessory.
Hybrids typically have larger electrical systems, and can more easily support electric power steering, electric water pumps, electric air conditioning, which are all more efficient than their conventional belt-driven varients.