Think about all the commercials you see on television. All those new products that companies want you to buy. Just about anything nowadays will have a money-back guarantee, and will say something along the lines of "no risk or obligation." They say that to make people feel safe and comfortable. Because if you have nothing to lose, well, why not try it?
The same idea can easily be applied to applications. Whether you're applying for a new job or just to volunteer somewhere, private applications are always going to make the person feel safer. They know that the application is only going to be viewed by a select group of individuals specifically responsible for choosing who will be brought on board. If they don't get selected, oh well. No one else knew they applied; they don't have to tell anyone; it all just disappears like nothing happened.
Public applications are a whole new beast. You're putting your interest out there for the entire community to see, the whole world even. This introduces new factors to the application process that can make those applying feel uneasy about the process:
Public criticism - not just about what's stated in the application, but about their behavior and actions in the past, their personality, anything people can chisel away at. Think of the process of running for President of United States. It's not a pleasant process. There's a lot of digging up dirt, a lot of lying, a lot of pretty much everything. Some people can stick it out and ignore it, some can't.
Unwanted attention - if you're like me, you don't like a lot of eyes gazing at you just because you threw your name in a hat. Putting yourself out there will generally increase the amount of traffic on your activity as others try to gauge out who you are and what you're like, which some people might find unnerving.
Uncontrolled association - when you submit an application privately, you have explicit control over exactly who you associate yourself with. Such as on a job application, you can choose which references you want to include and stick to ones that are going to contribute positively to your application. In the public context, anyone who knows about it can mention their association to you, which may not always be a good thing. Being associated to certain users or policies could sway the public's opinion about you
So yes, public applications will naturally knock off some of the possibilities as far as who is going to apply for the position. You have to ask yourself, though - Is that a bad thing? For a very public role such as being a moderator, these factors are probably very helpful in weeding out some of the applicants who might not be up to the task. Face it, moderators put up with a lot of crap. If they can't face the criticism and attention that comes with applying to be a moderator, how well are they going to be able to handle it when they actually are a moderator?