You've sort of got the answers and various additional colour and texture across all your responses so far, but some crucial points have only been hinted at, and a few responses could mislead the uninitiated (e.g. making it sound like lanthanum goes in the coatings, not the glass, further confused by mentioning fluorite, which goes in both). There's some interesting historical info which does inform an aspect of your question - why is LENS A worth a decent used car while LENS B is worth a crushed matchbox car, but you seem to want an answer independent of "this lens is boss because So Andso took Famous Photograph of Subject Performing Activity with it".
However disregarding things like the mechanical virtues of its housing, its system (i.e. what camera it meant you could use before everyone with an NC mill let you plonk it on a digicam), even the haptic (tactile feedback) of using it is a bit like asking "what's so good about Some Car Engine, but only tell me about its cylinder dimensions" (or whatever your own preferred non-optics analogy may be), because those are quite a substantial part of why So Andso had it on his or her camera in the first place.
I assume you were being at least a little facetious when you wrote:
Originally Posted by NickTrop
Often the lens make/focal length/widest ap is posted along with the picture.
but of course some people might not realise that quoting equipment and exposure settings used to be a requirement for publication/competition entry/exhibition (in the days before disaster voyeurism and the death of print journalism allowed the city's main broadsheet to print mobile phone photos on their front page *shake fist at cloud*). Inevitably seeing the same lens name appear in the details of lots of highly regarded photos will make it desirable, while one that's never associated with them withers into obscurity. As an entirely unbidden aside the staff of print journals of yesteryear cared very much what lens was used to capture their material, as it informed where, how large and with what crop they could print the photo ahead of final proofs: layout wasn't pure trial and error... Having distortions, aberrations etc. wasn't a disqualifier, but having unknown (or unknowable) ones often was.
There's also the Sir Pedant of Forumposting point that it isn't the widest aperture that's quoted, it's the focal ratio of the lens system. Since this is the Optics Theory forum: Lens systems incorporate a field stop (sometimes abbreviated "f. stop") to limit the propagation of rays which would otherwise not produce well formed spots (e.g. marginal rays whose counterparts don't reach the image plane) from contributing reflections and scatter inside the lens system. The adjustable iris field stop allows the trade off between aperture and off-axis ray selection to be fine tuned, but does not change the focal ratio (sometimes f/# or "f ratio") of the lens system, which in combination with the focal length of the system implies the degree of curvature of its surfaces (or optical power). This affects image quality independently of how big or small a field stop is installed/set, so they aren't the same thing (a 50/1.4 lens stopped down to f.16 works harder than a 50/16 lens to produce the same spot sizes).
The reason lenses are identified in this way is that the focal length divided by the focal ratio literally tells you the maximum aperture of the lens. I don't mean "1.4" or "2.8", I mean "80/4"=20. So an 80mm f4 lens has a maximum aperture of 20mm. This tells you the theoretical maximum resolving power of the lens (in this case the diameter of the first null of the Airy function is 30.5 microns, or put another way "the diffraction limited resolution of this lens is 30.5 microns"). A little pondering should convince you that smaller focal ratios give smaller theoretical spot sizes, and so are to be targeted, within the constraints of the imaging system. So, e.g. there's no point achieving the diffraction limit of this lens if its intended application is a Canikon DblaD with 5 micron pixels (which by the way are covered with a 2x2 Bayer filter so each image pixel is really 10 micron wide and a pixel per spot would be oversampling, so you need at least 2 pixels (3 is better, 5 is ideal) to make best use of your light so now 20-50 micron spots are the absolute limit of the image detector, though that would assume pixels have no gap between them, in reality they do). You can see that an economical compromise should be reached for any sensible lens manufacturing business.
[Edit: in the above paragraph I originally miscalculated 3 micron spot sizes as the optical limit of a 20mm aperture, which would have made a glaring mockery of lenses designed for dSLRs up to a few years ago assuming zero padding between pixels. I assumed the correct answer would consequently imply current 30ish megapixel sensors just horribly oversample any but the best and longest lenses (think $14k sports telephoto lightbuckets), but I see that the Canon 5D Mk IV still has 5.36 micron pixels (and indeed 36mm/6720 pixels on the long side is 5.357 microns per pixel, so it's the image pixels, not detector, which are the quoted size) and this makes me, if anything, more convinced that a lens which looks good on a dSLR can get away with simply targeting a spot size of 32 micron (4 pixels) across the image plane and be crowned King of Sharpness by the interwebs, but that requires only a decent aperture which is comparitively easy given sufficient glass. Try and get so close to the diffraction limit with a tiny, light 50 or 35 of yesteryear and the job is much harder, and I'm sure fine grain film will happily show you much smaller spot sizes than that if you give it a honking great 80/1.9 etc (thinking of things like CMS 20 or its period equivalents). Heeps of people put old film lenses on digital cameras. I've never seen a Canon EF lens-to-M645 body adapter... I wonder why?]
So, to finally bring all this together into what you actually wanted to know: LENS A and LENS B might have the same numbers (say: 50/2.8) but may have been built for different imagers (digital/film - what data pipeline if digital, what emulsion if film?) which inform what their target performance was. Then how well does each meet that target (or how far short do they fall)?
Fallacy 1. "All these primes have the same formula" has already been battered a little, let me crush it. A "6 element double Gauss" tells you this much about a lens: It has two positive elements, two negative ones either side of a field stop, then two more positive ones. That's it. It's like saying a car has "a V8 engine" or a pushbike has "18 speeds". Could be a F1 car or a Range Rover; a TdF winner or a supermarket Huffy, bla bla etc. pick your own alternative hobby. They certainly aren't all the same, and they aren't easy to get right even if you reverse engineer something to get you close. Thousandths of a millimeter on the curvature radii matter, ditto spacing, if cementing elements what cement, how much force etc. Then the glasses. There be not simply crown and flint glasses. The original U.S. Tessar patent has two different flints and two different crowns iirc. Schott catalogue has about 100 different glasses. Big companies can get exotic glasses (lanthanum, thorium, fluorite low dispersion glasses etc.) easier than smaller ones (or at all) and require large orders from Schott, Ohara et al to do so, which they pass on to lens/camera manufacturers (c.f. Rolleiflexes with Xenars because Schneider didn't have as high minimum order volumes as Zeiss). All of these need to be tuned to work together (the computing overhead already mentioned) and That Ain't Easy. I'm designing a "classic double gauss" lens system at the moment using COTS elements (fixing a lot of my geometry for me) and even that is quite hard to get right on axis (don't ask me about field of view, it's terrible!). All the good lenses and most of the merely adequate ones are unique in their geometries and glass compositions, as well as their coatings. The coatings allow freer geometries, but are variable in their quality and longevity. Combinations of glasses and coatings can prove to work well together or not so well, some immediately, others 50 years later and we sometimes see those effects in their market value. The relationship between companies has an effect: the coating firm that loves dealing with the optic grinder and the photographic lens manufacturer a) gets the best blanks of the top pick glass b) doesn't ship the dreg lenses from the outside edge of the coating chamber to their most valued client if they can help it. Was true then, is true now.
TL;DR - they aren't all the same formula. Yes, Leica (or whoever) did dope their glass, because all glass is doped.
Fallacy 2. "If you know all the dimensions and have the same glasses, you can make the same lens". Optics are specified to a surface tolerance in terms of design wavelengths. The terminology is "lambda on NUMBER" where NUMBER is typically 4 these days for general quality, and I doubt many photographic lenses are made with better surface tolerances. Lambda/20 sets you back a lot more than 5x the price, and is of limited availability, so guess whether Canon or Mark Bakovic has a better chance of getting elements ground to that tolerance. I'm talking here about lenses ground with NC grinding centres, but still working with essentially sandpaper and grit paste (ok, high grade stuff but still - mechanical grinding). Access to a fluid jet polishing machine can reduce overhead for lambda/10-/20 surfaces and enable even tighter tolerances, but you'd need a time machine to FJP an Elmar. So how does a Vaunted Lens Manufacturer c. 1950 get their LENS A to be so consistent and so consistently good (both keys: professionals break lenses, they also want their replacement from NYC to work the same way as the one they left in 12 pieces in a field in the Congo), while LENS B isn't (if they are indeed materially and by optical design the same)? Chucking out the elements that fail the interference test (test plate, fringes, lambda/NUMBER make sense now?). This might mean chucking out a lot of elements. Or selling them to Value Oriented Consumer Lens Manufacturer for use in LENS B...
TL;DR - the elements might be the same shape, the surfaces aren't.
Fallacy 3. "Modern lenses are perfectly corrected". Nup. Not even close a lot of the time, because you can just profile the aberration and code its correction into software with digital photography. Also see previous discussion of diffraction limits and pixel sampling (within the time limits imposed by The Res Wars). Also see other threads here about sharpness and photography technique c.f. old lenses and new (phase contrast autofocus is really quite capable: how many wave peaks of a green beam of light can you see through your split prism focusing screen?). Modern lenses are well tuned to their target application, which in some cases boils down to winning at Flickr, so really providing a good file to Lightroom is all the lens has to do. Even things like nanostructured coatings are available to give you maximum wallet/smugness transference without, y'know, being wise to schlep into a peat bog to get "The Shot". Don't assume a lens with aspheric elements (or better/worse yet, a *single* aspheric element) is automatically perfect. Many optics are "aspherized" i.e. a spherical ground optic with a cast (yes, from a mould) plastic cap to provide the conic surface. These are... less than highly reliable, so a large amount of QC fails would be the penalty. Besides, spherical aberration and spherochromatism are well corrected in almost all historically well regarded lenses, at least over their working wavelength range (hi there, 1920s Tessar! no I don't have any orthochromatic film for you today...) and badly in some that are highly sought after by the "soap bubble bokeh" brigade! Such a lens may well be very impressive, but aberrations exist on a spectrum, as do their corrections, and "perfect" correction of everything isn't really possible with a refractive system, even over a limited wavelength range.
TL;DR - all lenses are crap, to some degree. The trick is to make them crap in interesting ways.
So, to conclude:
Some have better designs. Some have better optical materials. Some match these two better than others. Some simply are more reliably able to achieve these goals. Almost none are highly regarded (or lowly) wholly without merit. As for specific examples: in LF Aero Ektars are legendary because they have a Don't Go To The EU-grade chunk of thorium doped glass providing a lot of optical power with very little dispersion => great lens (but keep it away from your kids). Pentax SMC (see earlier post, very interesting!) or Rollei/Zeiss HFT etc. are very good coatings, and coatings count for a lot with more elements (as discussed). Prewar Zeiss incorporated a glassworks that had access to great sand. Etc. etc. More or less answers your question (before I got here)
My point: They aren't all the same. They are all similar. If you printed a thousand photos shot with each, on different films, at different enlargements you'd probably see the difference. At 72 ppi after scanning to "bring out the... [whatever]" with no consistency in the equipment between the negative and your monitor? Well, no-one should expect to see that with a "good" lens (i.e. not a Meyer telescope for bubble hunting or an unfiltered 20s tessar on a CMOS digital sensor).
Reference: my time as an astrophotonics and laser physics student + misspent photography enthusiasm (ongoing).