But those who care about the details achieve truly high quality results overall. It extends to all areas of the design, not just to the parts you can't see.
In the movie "Who Framed Roger Rabbit," there's a scene in a dark room where Roger Rabbit (an animated character) flies across the room, knocks a hanging lamp around, and the lighting becomes so dynamic that all the shadows move around including the animated character's shadow. Here's the scene in question: http://www.youtube.com/watch?v=_EUPwsD64GI
This was such a small detail that it would have been forgivable if the animators had left it out entirely: if they had not moved the lamp, kept the shadow steady, no one would have really noticed the difference. It would have been 100 times easier to animate and the effect wouldn't really have been that different.
But they did it anyway. The term was later coined, and "bump the lamp" is used throughout Disney (and probably other organizations) to mean something akin to "go the extra mile"—but I see it as having a special significance to design.
You're right, most people won't notice. By that logic, you could cut corners a lot of other places too. You could be lax about button colors matching exactly, or per-pixel sharpness on the map and buttons. No one would probably notice.
But if you go for every detail like it was the most important detail, you have the possibility of reaching a level of design quality that is superlative, and some people will notice. Others will not notice directly, but will see that the piece exudes style and quality subconsciously, due to the attention to small details. If you carry this into other areas of your work—programming, customer service, market strategy, marketing, and more—then you have a chance to create something of true quality.
If you don't pay attention to detail at that level, well, you might have the chance to actually get something done. Yes, it's a balance, like everything else. But you have to know that it won't be quite as good, and understand that yes, you are sacrificing something, even if you can't see it.
The fact of the matter is that people usually do notice, if only subconsciously. If you show someone a clip from a Disney movie and a clip from a Hanna-Barbera cartoon, they will be able to tell that the animation quality is much crappier in the latter. They won't be able to tell you why, but on a visceral level they will be able to feel the difference. 
This visceral feeling of "quality" is what gives Apple's brand such cachet. Sure, their skeumorphism can be tacky, but it's never slapdash. Their apps always run at the highest framerates, and their touch interactions and animations have always been tweaked to a level that their competitors can't match.
Now, if you asked someone what the difference was between Mobile Safari on an iPad Mini and Chrome on a Nexus 7, no one would say "the scrolling is much smoother on the iPad, and the variance in framerate is much lower as well, which contributes to a feeling of responsiveness and playfulness", but those features still contribute dramatically to the "feeling" you get when using one of the two devices.
Now, sometimes it really doesn't matter. The example in the OP seems particularly silly. However, don't underestimate the value of "bumping the lamp" when it comes to design or interaction development. People will notice.
 The answer to the "why" has many parts, but mostly involves the fact that Disney animation runs at 24fps while HB cartoons ran at 12fps or even 6fps. Disney animation involves characters that are carefully articulated and that move realistically and with a sense of weight behind them. Also, the general quality of animation and camera work was just better -- look, there are a lot of reasons. They all add up to the difference between Scooby Doo and The Little Mermaid.
It sounds like they optimized for the wrong thing.
You can take your Hannah Barbara example as proof that some intangible things have an aesthetic impact, but you still can say when you see the two cartoons that one seems to be of better quality -- I hardly thing that is 'subconscious'. I think this article is intended to focus on the things designers do that are stylish yet unremarkable or wholly unnoticed, which any sensible designer must agree is something of a problem for web designers.
Now, it works because of the HB cartoons' light-hearted style and premises, but would seem tacky and ridiculous for a film or episode attempting something grander (i.e. a feature film like The Little Mermaid or Toy Story).
 For example, many HB characters don't have necks, or have extremely thick necklaces or collars where a neck would normally be. This allows you to swap out head and face animations with much greater freedom, and makes it harder to detect camera errors when the head and the body don't quite line up correctly. Is that bad? Not necessarily, but if you rely on too many of these tricks you end up painting yourself into a corner.
 Also, some HB characters don't even walk correctly. Their torsos and upper bodies stay perfectly motionless while their legs flail about beneath them. It looks cute and storybook-like in certain applications, but completely ridiculous in most others.
I mean, in terms of the 80/20 rule (the last 20% of quality costs 80% of the time), we're not talking about turning an 80% thing (i.e. typical in-house enterprise software) into a 90% thing here. Hell, even going for 95% is something I believe we should all go for, and fight our colleagues, bosses and project managers, and customers for. But the OP's example is more like turning 99.5% into 99.8%.
Personally, I believe that much of the value created lies in the final 20% that takes 80% of the time. If what you say is true, near anyone can create something that's 80% of the way there. It's your job to get as close to 100% as you can realistically, while balancing time and money and value. It's not an argument for unlimited perfectionism; it's an argument for quality. The higher you can get, the better you will produce, and the more you will succeed. And don't just limit this to product quality; extend it to everything you do (including customer service, including planning and resource management, including hiring and culture) and you'll create a company that has a chance of being better than the others.
Perfect? No. But if you're not striving for creating something of quality (in other words, of value) then what are you doing, exactly?
Edit: There's a caveat here, I think. It may not be your job to worry about every design detail down to the pixel. It doesn't have to be. You can find someone who can help you produce what you need at the level of quality you desire (like the designer the OP is talking about, presumably) and you can worry about all the little details of running your company. Details that I'm guessing a designer might not understand or care about, but you do.
Maybe for this case in isolation going the extra mile to this level of perfection is not justifiable, but if the designer is doing good work and meeting deadlines then you should keep your bean-counting micro-managers far far away. To attempt to quantify everything (even in programming) is to take the human element out of creation, and that runs a severe risk of throwing the baby out with the bathwater.
There are coders out there who strive for absolute, unlimited technical perfection. They rarely if ever ship - and ditto for designers in that boat.
In reality most designers, and programmers, walk a balance between the bottom-line functionality of their work and less concrete (and more subjective) qualifiers like code quality or sub-pixel scaling.
There are many analogues between design and code here. Most people here would agree that sloppy coding standards aren't deadly, but encourage sloppiness on a larger scale over time. Ditto, sloppy design is insignificant in the individual case, but have the tendency to accumulate and lower the bar.
The example OP chose was particularly bad, mostly because as an iOS dev I know for a fact that getting 1x assets right doesn't take that much time. The usability improvement over time invested is not atrocious - certainly nobody spent an extra week just to get some pixels to line up just right.
The major problem with scaling 2x assets down to 1x devices is that thin features become muddy and blurry, while code-wise UI elements may become aligned against half-pixels, resulting in filtering that produces blurriness. Both the design and code elements of this are easy fixes for any competent iOS developer/designer duo, and takes no more than an hour or two at the extreme most. You get a pretty clear legibility improvement for not a lot of man hours.
Not to mention the very noticeable blurriness of the buttons in the "unfixed" versions - things that users do notice. So much so that some of them can actually articulate it (as opposed to merely being a subconscious annoyance).
Design is not art in and of itself, but you can't talk about one, without talking about the other.
If you are not in that situation, could it be because you weren't perfectionist enough in the past?
E.g Nokia vs Apple.
And tried polishing symbian. Which had a number of terrible design decisions at the core (which may or may not have been good ones 15 years ago, when they were introduced into EPOC, the ancestor of Symbian). And those decisions affected everyone who worked with it, both internal Nokia developers and external app developers. It made developing for Symbian a pain. It was natural, that iOS and android (and Windows) passed by.
To add what little I can to this, the reason that drives me to go to these lengths is the craft. Design, like code, is a craft. A good craftsman just does not leave mistakes in their work purposely. Whenever I do this it slowly drives me crazy until I either go back and fix it, or drop the project entirely (if there are a lot of mistakes and/or things I don't really like), usually citing some reason like 'it's too messy' or 'i lost interest'.
For me, design and programming are not all about a/b testing, 100% time efficiency, etc. A lot of it is about art -- making something you know is great and shipping it. When you know something is wrong with your work as an artist, it will bother you until it's fixed.
I am a lead designer, and I know that not every piece needs to be a masterwork, but ever piece of software does need to ship.
Absolutely not the case. We're talking about quality specifically, and why someone might choose to pay attention to finer details versus ignoring them.
Your reasons for ignoring details might be perfectly valid and entirely appropriate. The project may not call for artistry and mastery. All very good points for a different discussion.
Don't think for a moment that those who believe high quality is an important factor are implying that it is absolutely necessary. That is simply a false dilemma.
My point was that the ONLY thing that was listed was details being the separating point, and that there is more to it then that. It wasn't meant as a quip just a clarification, if anything a question even, not requiring a snarky answer but maybe a genuine one. But, I forget, I'm on the internet and snarky is everyone's default for whatever reason.
Basically "I completely agree with you, but wish to do so in a humorous and sarcastic manner." Apologies.
The same sort of technical ugliness and lack of optimization happens in plenty of programs.
In theory, one should never optimise prematurely, and get the application working first. Only at the end profile the speed, find the one single slow loop or algorithm, and optimise that.
But plenty of applications acquire slowness not due to one core slow algorithm, but a lack of concern for even trivial optimisations everywhere. You end up with something like Eclipse where a keypress can take multiple seconds to appear, or a "cancel" button might take minutes to stop whatever task. And there is no place to optimise, because the inefficiencies are spread out by tiny bits everywhere.
1. A work of art is created by definition to be beautiful. While a beautiful UI is awesome and definitely something to strive for, it is not the end goal.
2. A UI can be changed after its release. Unlike art, you can release a UI and then later change details that are important.
3. Most companies do not have the manpower or time to actually get to the place where the entire product is pixel-perfect. pg's mantra is: "release early." This means that not everything is going to be perfect and you have to decide what corners can be cut and which cannot. (You cannot simply state, I am going to cut NO corners).
4. A UI is likely to change. When you first release a UI you dont really know how it is going to be used. Therefore, as the OP suggests the improvement from 95%-98% really does seem like a pre-optimization.
That said, the UI should make the experience easier in that it aids the function. I would take it a little beyond that where you want something attractive to look at, but it doesnt necessarily need to be in the v1. As the UI matures over time beauty in design becomes a more important goal.
Why is the always treated as truth whenever non-artists discuss art? This is a backwards, 19th century era concept. Art broke out of this generalization over a hundred years ago.