What does a surge protector do that a breaker doesn't?
Surge protectors and circuit breakers both cut off power during exceptional conditions, but they have different purposes and react to different events:

Circuit breakers

Circuit breakers have only one job: to prevent the wiring inside your house from catching on fire. That's it, they do nothing else. They don't protect you from shocking yourself, or from lightning, or from your cat chewing through a power cord.

Circuit breakers work by monitoring the current (number of electrons) flowing through the wiring, and cutting off the power if the current is too high for too long (which puts the wiring in your house at risk of overheating). The higher the current, the quicker a breaker will trip: it may be just a split second or it might be hours. The idea is that a transient burst of power (e.g. when your fridge turns on) is not particularly dangerous.

The rating of a breaker depends only on the size and type of wire used in your house (which in turn is based on the expected usage). Typical sizes would be 15 or 20 amps for outlets and standard lighting, and higher, dedicated breakers for high-use appliances like stoves or hot water heaters.

Surge Protectors

Surge protectors have a very different goal: to quickly disconnect equipment from the power supply if there is a power surge. Surge protectors don't monitor the quantity of electricity, but rather it's voltage. An unexpectedly high voltage can force power into appliances beyond their specifications. Typically surge protectors cut off power when the voltage reaches 300V or 400V, which may be indicative of a lightning strike.

Surge protectors also have the job of absorbing energy due to a voltage spike. 

Neither surge protectors nor circuit breakers do anything to regular minor fluctuations in voltage like a brownout.

In my opinion the need for surge protectors is somewhat overblown, since harmful power surges are relatively rare and there's no guarantee that a surge protector will be robust enough to prevent damage anyways. But they are convenient if you need extra outlets.

EDIT
@Ecnerwal's answer is correct that technically surge protectors don't disconnect the power, they redirect (or absorb) it. And his answer has much more technical detail about the types and purposes of surge protectors.