There are two different types of "chief resident". One is where they just choose a 3rd year resident to deal with all the admin tasks that the P.D. doesn't want to deal with. The 2nd type of "chief resident" is where someone devotes an entire extra year of their life to being a sort of mini-attending who staffs the residents, deals with all the open globes, while also managing their schedule. This latter type of chief resident is more prevalent among bigger programs.
Both come with the benefit of harboring good/better relationships with the faculty, which is rewarded with a great LOR. Many people do it to improve their C.V. before a competitive match (e.g. Oculoplastics or Retina). I guess the third type of "chief resident" is where someone goes off and does their fellowship, and then comes back to their original residency (e.g. Wilmer) to serve an entire year as that mini-attending.
I think the only way I would consider doing one myself is if I was staying in academics or thought I couldn't match in a competitive fellowship. Otherwise, it is a year of lost income with very little gain in your surgical skills.