Fame
Game wardens, like most law enforcement officers, place public safety and order above the pursuit of personal recognition. That's a good thing, because even if they wanted it, they're probably not going to get it. The closest a game warden gets to fame is when a wild cat escapes from the local zoo and they have a seven-second sound bite on the 6:00PM news. You may also become moderately famous with various Boy and Girl Scout troops in your area, but that's about it.
The most famous game warden was Guy Bradley, and not for great reasons. Guy was a Florida game warden in the early 1900s (which you could probably guess based on an old-timer name like Guy) (source). In 1905, Bradley approached two men he suspected of poaching birds in the Everglades.
The two men, who were obviously doing nothing wrong (not), proved it by shooting and killing Officer Bradley. His death made national news and galvanized conservationists across America—including noted nature-lover President Theodore "Teddy Bear" Roosevelt—and Bradley was dubbed Bird Protection's First Martyr (source).
In short, nobody wants Guy Bradley-style fame, and there aren't many other ways of attracting national attention as a game warden.