What benefits is there to using a PEEK implant for cheek bone/ infra orbital augmentation over the the typical silicone?

Peek implants are firm. They require more dissection, screws to secure (usually), and are less forgiving. Silicone implants are much preferred, unless, for some reason, you must have a custom made implant (such as due to prior injury or birth shape issue)

Compared to silicone implants, PEEK implants are considered more biocompatible and lower the risk of infection, inflammation, and rejection.
Silicone implants are cheaper than PEEK implants but are more prone to complications, such as infection and migration, and may require replacement over time.
There are other biocompatible materials like Polyethylene which I would favor for custom facial implants.

When it come to choice of material for any form of aesthetic facial augmentation, there are numerous factors to consider. While many people getting focused on the purported biologic differences between the materials (there isn't any really as an implant is an implant from the body's perspective...they are all foreign materials), the more important factors to consider are ease of insertion/removal and cost. It is in these issues that the significant differences between the various materials become apparent. In the U.S. there is going to be a major cost difference between PEEK and silicone for custom implant manufacture. Silicone has been and will always be a material that has been made for the cosmetic surgery market which has always been cost sensitive. A material like PEEK has its origins in reconstructive surgery market where insurance reimbursements occur and the cost of implants for those cases, where neither the surgeon or the patient is paying for it, is considerably higher by. factor of 3X to 4X. Due to U.S. las a manufacturer of a medical device can not adjust their price based on who there customer is.....the cost to the reconstructive and cosmetic patient must be the same.

There are various materials used for cheekbone augmentation. PEEK, Silastic, Goretex, Medpor etc are examples you may see. The type of material chosen can depend upon the 'look' you prefer. Silastic is widely manufactured by many companies and the cost can be a factor compared to others. The
'hardness' of the look eg male versus female; the degree of 'fullness'; the angularity desired etc are all considerations.