It's general knowledge in typography that Helvetica is the most legible typeface.
That's only true at very large sizes—say 5% of your total field of view or larger—and it is IMO highly debatable even at those sizes.
At small sizes, particularly for people whose vision is less than perfect, Helvetica Neue makes Comic Sans look readable by comparison. It's not a question of the screen's resolution; no matter how precisely you render two letters that are separated by a distance that's less than your eye's circle of confusion, you still can't distinguish the strokes from one another.
For example, on my brand new MacBook Pro with retina display, I have no trouble whatsoever reading Courier New at 11 point. It is easily readable, and every letter is visually distinct. Same goes for any number of other fonts, including the venerable Lucida Grande. On that same hardware, my eyes struggle with Helvetica Neue even at 18 point, which means if I want it to be readable, I would get substantially less content on the screen even when comparing it with a fixed-width, serif font!
And the reason for the readability problems are a decided lack of legibility in Helvetica Neue. With Helvetica Neue 12 point, when I look at the word "pill", the "p" touches the "i" until I'm six inches from the screen. And depending on where the letter happens to fall, it may or may not be possible to tell the difference between "pom" (the juice) and "porn" (naughty stuff on the Internet) without getting ridiculously close to the screen. Sometimes the gap is visible, sometimes it isn't. In other words, the tracking is simply way, way, way too tight to qualify as legible. Remember that when designers use Helvetica, they painstakingly tweak the kerning to ensure readability at the target output size. As a general display font without that level of hand-tweaking, Helvetica and Helvetica Neue are crap.
But Helvetica Neue's problem goes way beyond over-tight tracking. The most critical requirement for a font to qualify as "legible" is that you must be able to distinguish letters from one another. Helvetica Neue fails miserably at this, though not quite as badly as Helvetica or Arial.
For example, look at a lowercase "L" and a lowercase "i" in almost any font, and you'll see that they are decidedly different heights. This is deliberate; it makes it possible to tell the difference between a pillow and a plllow, (which I believe is Ancient Egyption for an unreadable typeface, but I could be wrong).
Not in Helvetica Neue. They're the exact same height. This makes it excessively hard to read text that combines those two letters, particularly at small point sizes where the gap in the lowercase "I" is often hard to see.
And speaking of "I", is that a capital "i" or a lowercase "L"? If you're reading this in Slashdot's default font (Arial) or in Helvetica or Helvetica Neue, you probably can't be certain, because the two letters are nearly indistinguishable. So when I say I'm "Ill", do I mean that I'm sick, or that I'm three years old in Roman numerals? At 13 point, even on a Retina display, a capital "i" and a lowercase "L" can look literally identical, depending on where the letters happen to fall and how font smoothing interacts with them. And that's even with getting my corrected-to-20/20-vision eyes as close as a couple of inches from the screen.
Legible, my ass.