Pros:

• Many small strobes and studio lights already have optical triggers built in, so you may already have them.
• Because optical triggers are often built in, this reduces the number of items you need to carry, as well as the number of interconnects. Fewer parts leaves less room for failure or error.
• Optical triggers are inexpensive if you do need to add or replace them.
• Many optical triggers are built in or powered from the strobe's triggering circuit, so you don't need extra batteries.
• Optical triggers (visible light, rather than infrared) just detect the burst of light, so you don't need a system made up of compatible triggers from the same manufacturer. You can mix and match.
• If you travel, optical triggers can be used anywhere. Radio triggers operate on licensed frequencies which do vary from country to country.
• They can work reliably in a studio environment where conditions can be controlled.

Cons:

All these cons boil down to the issue of the reliability of detecting the triggering flash or detecting something other than the triggering flash.

• Optical triggers can fail in bright sunlight, as there's not enough contrast between the sun and the triggering flash.
• Optical triggers are limited in the distance the trigger must be from the triggering flash - the flash needs to be bright enough by the time it hits the trigger.
• The triggering flash may be set to a power level for creative purposes that is too low for the trigger to detect. This can force you to change your lighting scheme.
• Any object between the triggering flash and the optical trigger can prevent the flash from being detected. This includes modifiers (umbrellas, softboxes etc), flags, the subject, and solid objects like walls. (When shooting interiors, it's common to hide a light on the other side of a doorway or behind a sofa, for example.)
• An optical trigger usually needs to be facing the direction of the triggering flash, even if there are no blocking objects in between. Otherwise you're relying on reflected light, which may not be powerful enough to trigger the strobe. This can limit light placement options.
• Optical triggers are prone to false-positives and can flash with e.g. the reflected light from a passing car. This can cost battery life and cause you to wait for the strobe to recycle.
• Optical triggers can be falsely triggered by other people's flashes. If working alongside other photographers, or just members of the public with point-and-shoots, any flash will trigger your lights. If this is happening, you have no options other than to eliminate the source of the flashing or move away.
• If any of your equipment issues a pre-flash (used variously for auto-focus assist, red eye reduction etc) an optical trigger will fire before the main flash, often not leaving the triggered light enough time to recycle to flash again for the shutter. The result in your photo will be as if the optically-triggered lights did not fire.

It's for reliability reasons that many people choose radio triggers, even inside the studio. Those are not without their problems (most pros for optical triggers are cons for radio), but I think those are out of scope for this question.

Optical triggers are a good, less expensive solution if you can either control the environment in which they're used, or are happy to deal with a few reliability issues.