I am generally interested what are the benefits of Putnam (from both students and Professors' point of view), apart from the fact it is fun (for some) to solve difficult maths problem. I recently read a post on here that a good Putnam score would aid PhD application. If someone would elaborate on this a little, it'd be great.
The genuine benefits I think all fall in the category of "distinguishing oneself" ... from other applicants to grad school, for example... much as with the IMO (Int'l Math Olympiad).
That is, an otherwise merely-very-good record (out of 100s?!?) can distinguish itself from others by a pretty-darn-good Putnam score. What does it "really" mean? Well, maybe not so much about higher-level mathematics, but the "exclusivity" of "having pretty darn good Putnam score" is undeniable. It's a thing that can be capsulized and "explained" to the public in PR, whereas it's harder to popularly explain progress or work in genuine mathematics (without severe distortions...)
Even at beginning-undergrad levels, kids who do at-all-well on the Putnam get substantial recognition within their math dept, often are treated more forgivingly of other quirks or inconsistent performances in routine coursework. (Given the stodginess of much of the routine undergrad curriculum, this is a good thing, since it is completely reasonable to not quite be able to consistently comply with the apparent requirements...)
The flip side is that there's a hazard here, as in all contest-math, that the quick gratification (if one receives it...) may make long-term study and research look less rewarding in comparison. In happy cases, one can "warm" to more prolonged projects, and more-delayed gratification. But quick contest-things like Putnam can be encouraging, and are noticeable pluses on a CV or grad school application. Just should not be over-interpreted.