My opinion is it doesn't matter much if you're not interested in an academic career. It looks good on a CV, but if you're going private practice, I don't think it makes a huge difference. The job market isn't at a point where decisions typically need to go to that sort of thing as far as I can tell.
If I was hiring someone, I'd be more interested in the patient population and conditions they treated most in residency than their pedigree. It could be argued that the big names have a lot of resources and thus more opportunities to learn, but there is also something to be said for learning to treat patients on more of a shoestring budget so to speak.
I would say that the big name programs may be more likely to have specialty clinics or niche experts at their programs. If one is interested in learning a lot about that certain thing, then a big name place would have an advantage. I feel that may be more of a program specific thing than necessarily a big name vs lesser named program thing though.
In terms of fellowships? Well, in terms of mohs and dermpath, it may be a bit of an advantage moreso to be able to stay where you are for training but also in terms of getting good letters from big names and so forth. However, I don't perceive a huge differential advantage there. As you move along in training it becomes more personal and more of a "This person will make a great mohs/dp doc and I would totally enjoy working with them for a year" opposed to putting more weight on 'pedigree'. Expressed a different way, I'd rather work with and train a person from a lesser known program who is a nice person that I'd enjoy working with than a person from a big name place who my personality doesn't mesh with at all.
So it's almost like interviewing for fellowship is like interviewing for a job; the interviewer cares more about the quality of the person overall than just the name of the place they trained. That maybe doesn't always apply, but that's how things work from my perspective.
