Rolling around the Internet recently led me to this post by James Young (owner of Ten Foot Polemic) on how he uses skill checks in his OSR games. Aside from other points which I agree with by and large, he mentions that he avoids using ability checks because PC ability scores are pretty static in his system of choice (LotFP), so someone who rolled up 12 Strength during character creation would have the same chance of passing a given Strength check regardless of whether they were first level or tenth level. I agree that that is true, but I don't consider it a problem personally because I think it's entirely reasonable for certain tasks based primarily on raw capabilities rather than any learnable experience to have a chance of success independent of the character's level (i.e. I think it makes sense for tasks which would fall outside of what I'd expect to be modeled in a skill subsystem to depend on basic ability scores rather than being something which can improve over levels).
LotFP has at least one instance of agreeing with me on this point, incidentally, since opening stuck doors in that game system is a check that works just like a Strength-modified skill check except that there is no skill associated with it, but I digress.
Arguing opinions is rather boring, though, especially when the other person in the argument probably has no idea that I exist. Instead, I'd like to take this as an opportunity to muse out loud on some of the relative benefits and/or drawbacks of each approach, along with my expectations of how using one or the other exclusively would feel. Disclaimer that I have not actually played LotFP (despite it being one of the two OSR systems which impressed me enough to actually buy a hardcopy of the rules with the intention of using it some day) nor any other system using the same sort of skill system, so I will be relying on imagination and extrapolation for that side of the discussion. I'll try to keep it in good faith as much as I can, but I'm sure there'll be a difference between my imaginings and reality.
In the interest of being explicit about assumptions, when it comes to this post's discussion of ability checks, I'm referring to a "roll under stat"-approach, not a "roll + stat modifier vs. target number"-approach. I've used both sorts of systems in the past, and as evidenced by my using the former approach in my heartbreaker's rules, I prefer the former. Also, for the purposes of general applicability, I'm going to assume a d20 roll for ability checks, even though I prefer 2d10 for reasons of altering the distribution of results, as I've discussed previously.
Decoupling skills from ability scores has a few obvious impacts that I like. First, it allows for finer differentiation of characters. Just because two PCs have 12 Strength doesn't mean that they should have similar chances of climbing a wall, for instance. This can lend mechanical support to creating characters who're just unexpectedly good at something (such as an idiot savant inventor or an iron-thewed warrior with a knack for moving with the stealthiness of a hunting cat). Second, the greater jumps in the odds of succeeding for each skill point (or situational modifier) from working on a 1-6 scale instead of a 3-18 scale creates a stronger feedback loop with the human brain's naturally clumsy sense of statistics. Third, since the aforementioned decrease in the scale range implies a higher threshold to qualify for situational modifiers, it should tend to reduce the overall amount of consideration given to accounting for situational modifiers, which should streamline gameplay. Fourth, having a target number resolved on a single d6 roll requires less mental effort than having a target number resolved on a d20 roll (further improved by the lower prevalence of situational modifiers, per the previous point), which gives closer-to-instantaneous feedback to the player on whether or not their roll succeeded.
Within reason, greater differentiation of characters is a case for increasing complexity that I tend to support. While a lot of that can be handled through roleplay, it's nice to have the game's mechanics support what's supposed to be special about your character in an explicit way. On a related note, this can also encourage the player to think about why their character has certain skills, especially if their skills are selected through random rolls rather than free choice. While it's easy to say your character is skilled in Stealth and First Aid because you think those would be useful skills, it can be more interesting to be told that your character is skilled in Animal Handling and Sleight of Hand and be left to reason out why that might be the case (this can be taken a step further by looking at the party as a whole, too, i.e. if this group of four people has three who're skilled in Languages and nobody who's skilled in Bushcraft, maybe that means they're part of some academic institution that's based in urban areas or that they met each other while delving in libraries). In this way, mechanics and roleplay can inform each other in a manner that wouldn't happen when relying strictly on ability checks (using ability checks with modifiers for skill investment muddies this up, but I think the general point still remains).
The human brain is rather notorious for being ill-suited to grasping statistics, probably in part due to past evolutionary pressures that favored both pattern recognition and risk aversion heavily. 5% increments (as expected with a d20-based mechanic) are too fine to carry palpable impact. For most people, something having a 50% chance of success doesn't feel intuitively different from something having a 60% chance of success when they're just looking at the numbers, but they often can feel the difference when experiencing the results (as they should, since the latter is actually 20% more likely to succeed, despite the rate of success having only a 10 percentage-point difference!). Using a d6 resolution bumps up the increment step size to 17%, which carries enough extra weight that it'll tend to trigger the pattern recognition elements of the brain with respect to associating how good a character is at a certain task. Similarly, because our minds have an easier time of grasping the odds on a d6 roll, succeeding on a 2-in-6 skill check can induce more of a gambler's high than succeeding on a 7 Strength ability check, even though the actual odds are almost identical (33% vs. 35%). This increase in the impact of each incremental step also makes it easier for a GM to judge whether to apply any situational modifiers, since the intuitive thresholds to qualify for a +/-17% change will be clearer than a +/-5% change and will tend to result in using situational modifiers far less often on the whole.
Any time dice are being rolled, math is being done, and success or failure is being determined is time where actually playing the game is on hold. While it may be a difference of only one or two seconds on a single roll (or perhaps a total difference of one or two minutes, taken in total over the course of a whole session), every instant of delay between seeing the number on the dice roll and feeling success or failure compounds into a rapidly degrading sense of how much that roll is influencing what happens. Working with a result from a d6 roll (and one that is often lacking any further modifiers) is objectively faster than working with a result from a d20 roll, and if the GM wants to use the difference between the roll result and the target number to measure the margin of success/failure, calculating that is also faster when working with a tight range of exclusively single-digit numbers. It may not seem like a big difference since we are talking about fractions of a second in either case, but it is absolutely palpable in practice. If you're going to use a slower resolution mechanic, you had better be buying significant benefits with your sacrificed time for it to be a worthwhile trade, and that's usually not going to be happening with a simple pass/fail check.
Well, that all seems like a slam dunk in favor of d6 skill checks, right? Not so fast.
Though using decoupled skills can model the idiot savant inventor better than simple ability checks, ability scores being graded on a 3-18 scale rather than a 1-6 scale gives more granularity between characters of similar overall skill. While the 16 Intelligence inventor and the 17 Intelligence inventor are likely both far better at inventing than the average character, the 17 Intelligence character is also significantly better than the 16 Intelligence character (85% vs. 80% chance to succeed, or to look at it in another fashion, 25% less likely to fail), whereas approximating the same success rates on a d6 scale would give each 5-in-6 (whether that would be their actual check or not under a d6 skill system would depend on exactly what game system you're using and how much they might've invested in improving their checks, but the specifics are not as meaningful as the fact that their odds would be unrelated to their actual Intelligence scores). Thus, while certain characters can be easier to model with decoupled skills, modeling the finer differences between similar characters can be easier with ability checks.
While a d6 skill system is certainly faster to resolve and has certain advantages from its coarser granularity, I don't think those benefits offset how it tends to result in the actual numbers of your ability scores being meaningless. Assuming a non-uniform ability modifier scale (e.g. assigning ability modifier based on standard deviations from the average rather than [Ability Score - 10]/2), most OSR systems have no difference in the modifiers between 9 Strength and 11 Strength, for instance. However, one is supposed to be a character who's slightly weaker than average while the other is just on the upper end of average Strength. I think that should result in having enough of a difference between the characters to affect their ability to climb a smooth-walled chimney or swim against a strong current (this is particularly pronounced with a 2d10 or a 3d6 roll, where it's a difference of 36% vs. 55% or 38% vs. 63%, respectively, but I digress). If we're not going to use the actual ability score numbers for anything, there's no point in writing them down in the first place, particularly if your game system of choice has no expectations for improving the ability scores over time; you should do the 3d6 roll but just write the associated modifiers.
Now, there is actually some merit in relying on clustered modifiers instead of the individual ability scores. Namely, it reduces the impact of randomness while rolling for ability scores, thereby reducing the likelihood of a player ending up with a handicapped character due to bad luck during character creation. I can accept that as a design goal. It leaves me wondering why players would still be expected to track their specific individual ability scores as anything but an unnecessary nod to tradition, but it's nonetheless an understandable goal. I'd rather embrace the finer granularity of differences in general, but this is something that comes down to personal preference.
Speaking of preferences, I think the ultimate choice on which approach to favor comes down to how you like to characterize competencies. Using ability checks results in characters being able to apply themselves to a broad range of areas while tending to stick with similar approaches. As long as the player can come up with a way of utilizing their strengths (whether those are Strength, Intelligence, Charisma, etc. will influence the actual methods used, of course), they can apply those strengths to all sorts of different situations. On the other hand, using skill results results in characters being able to demonstrate a broad range of approaches related to a certain skill category. As long as the player can think of a way to apply their areas of expertise (whether those are Bushcraft, First Aid, Tinkering, etc. will influence the actual methods used, of course), they can apply their expertise to all sorts of different situations. Both approaches can foster creative application of lateral thinking, and it comes down to whether one prefers a higher baseline competence that rarely improves or a poor initial competency that can develop significantly at each level gained.
Having thought this through, it's actually kind of funny that both LotFP and I went with the approaches which seem the most opposite to the rest of the system. LotFP characters have pretty flat power growth through their levels, mostly just improving in their one core area (fighters gain attack bonus, specialists gain skill points, magic-users and clerics gain stronger spells that can be used more frequently, and non-humans gain in their signature skills [plus HP for dwarves and spells for elves] ) while having a solid baseline all-around aside from being bad at skills that they have no points invested in. In my game system, characters have significant power jumps at a handful of points over the course of their level progressions, affecting both combat and general capabilities, aside from ability scores staying stagnant by default. I intend to make an update to my system at some point to better match the mechanics to the feel and playstyle that I like (plus making some updates based on exposure to more ideas for handling various mechanics), so this is an area where I'll need to do some experimenting and introspection to see if I'm really using the ideal approach.