Some thoughts.
I know this thread is old, but I came across it before taking my faulty iPad to the genius bar yesterday, and was quite worried.
However, when I arrived, I was pleased to discover I was seen by someone I actually sort of knew (we both interviewed for Apple at the same time and were in the same "team" during the interview process), who remembered me, and provided me with a lot of insight into the refurbishment process, which really eased my mind.
Basically, he explained it to me like this.
Newer Apple products (iPads, iPhones, Post "Classic" iPods, newer MacBooks and iMacs) have moved away from easily serviceable and replaceable "off the shelf" parts to a "Glued and Soldered" model which are actually quite tough to get at.
As a result, they have to test much more thoroughly in the factory than they used to, and than most other companies do, and the failure rate (outside of accidental damage) are actually "very low" compared to most in the industry, and that the iPad and iPhone both the best in their sectors in terms of component failures.
The reason, he said, is that so much is tested before being sealed because they cannot be easily opened once the digitizer is glued into place.
Nonetheless, failures do happen, and because of these difficulties, they replace with refurbs rather than try to repair them because "They couldn't get inside and do it in store, even if they wanted to".
Basically, when you take your faulty iPad into the genius bar, they will diagnose the issue, then ship it back to Foxconn or the other assembly line companies.
They strip it back to it's basic components, and reassemble it into a working iPad (or iPhone, iPod, whatever). They always get rid of the component listed as faulty by the genius (eg, if it's an error with the headphone jack, that cable/board is gone, no questions asked), and never reuse anything considered consumable (eg, the battery), or which has been exposed to the end user (eg, the digitizer/case), or anything that is adhered to them (Eg, Antennas).
These components will always be brand new.
After that, your refurbished iPad will be a mix of recycled parts from multiple iPads (They do not generally repair a single iPad, but remanufacture a new one from lots of different parts). Basically, when an iPad arrives, it is stripped down into working parts, each one tested independently, once it passes QA is then placed into an assembly line to be built as new iPads would be.
He said the LCD is usually new, as so few of them pass QA (they are either damaged during the digitizer removal, or else have dead pixels, or discoloration) especially during the earliest batches of refurbs to come in, and the logic board may be new. He said that although he thinks they do recycle logic boards, there are very strict usage thresholds which they must not have passed. He didn't know specifically what they were, but said he would be "very surprised" if a recycled logic board would have been in use for more than 3 months, or used in a heavy way. Basically, because they need to guarantee them for the remainder of the warranty (or 1 year if sold) they don't like to use older recycled logic boards, and it is often cheaper to use new ones than take the risk.
He also added that as most of their replacements happen within the first few months of ownership anyway, it's rare that any of the components are going to be that old.
After that, everything else is likely to have been in another iPad before, but again, is fully tested, moreso than brand new components. He also added that while the core parts (eg the ribbon cables) of certain components (such as buttons and switches) will be reused, the actual parts you touch and interact with will always be brand new.
He said that, without opening it up or checking the serial number, it would be impossible, even for a genius, to tell the difference between a new and refurbished unit. He said that he sees "far less" refurbished devices coming back under warranty than new ones, and where they do, it is "almost always" because of software issues not hardware ones.
He says that they quite regularly swap out devices, even when they believe it's software related, especially when the issue is intermittent and not easily reproducible because "it's what the customer expects from us". He said that 9 times out of 10, when a refurb comes back, it has exactly the same issues as the device it replaced, because they have restored from the same backup. He said other faults with refurbished models are "rare".
He personally believes that refurbs are of better quality and reliability than new ones.
I got a refurbished iPad, in exchange for my faulty one, and have to say, I am very impressed. In general, I would say that it, subjectively, feels of better overall quality than the one it replaced when new. Everything feels "tighter" and has better fit and finish. Eg, my old home button was never really flush to the digitizer, and had a little raised edge, and a bit of play in it, which this one doesn't. Also, the digitizer itself is much more flush around the edges. Also, the LCD doesn't have the purple fringes at the edge that the old one had.
All in all, I would have no hesitation in taking a refurbished model in the future, after all, it's covered by the exact same warranty if it does fail.