In COMBAT, nat20 always hits - but if you needed a 20 or more to hit, then the nat20 isn't a "crit", it's just a hit. A nat1 always misses, even if you're +10 and attacking an AC 8 zombie. In either case, you (player) or I (DM) can and should describe why you succeeded or failed despite all odds to the contrary; a lucky strike to a joint, or slipping in the mud as you lunge toward the foe. IMO, these are moments to add "flavor" and "cinema" to the story.
In the past, I have tried house rules for Critical (and "Exceptional", i.e. "hit by 10+") hits that emphasized the nature of the attack. Like a firebolt or a lit torch igniting the target, a blade causing a bleeding wound, thunderclap deafening the target, or a warhammer knocking the foe back or prone. In theory, it was cool - which "d8 one-handed martial weapon" you took, or which cantrip you blasted with mattered a little bit. In practice, it was one more set of rules to remember - great for a computer game, rough at the table. So I simply made the batch of ideas "available" if the player wanted to sacrifice the die roll of the critical hit to implement a special effect. [Our crits are the "crunchy crits", "max roll + actual damage roll + bonuses"... so your dagger strike would do 4+d4+DEX, but you could drop the "d4" part to cause a bleeding wound, or perhaps give your ally Advantage to hit as if you were flanking. The stronger the effect you tried for, the greater the chance the target saves against it.] Now 5e24 has Weapon Masteries, which incorporate most of this, so mostly it falls by the wayside. I might still have a crit inflict an extra effect - PC or monster! - just to switch things up, but that fiat is used sparingly.
Note that while weapon crits do "crunchy crits" in my game, spell crits are a little different. If it's an attack-roll-spell, you just roll double damage; this still gives martials a higher average damage. BUT, if it's a SAVE-spell, a nat1 on the target's save indicates they take max damage from the spell. Last session, the celestial warlock tried to Intimidate a rival adventuring group by dropping a wall of fire on their wounded friend - intending to revivify him the next round if necessary. The hapless ranger rolled a nat1 on his save, though, so instead of taking maybe 22pts and bleeding out, he took the full maximum 40 and was incinerated! Whoops!
(One thing I will not do, though, is have "fumble" or "crit-fail" results in combat. Yes casters now make attack rolls too, but martials are still making many more rolls (Extra Attack, TWF, etc.), and there is no need to saddle them with extra mishaps!)
In SKILLS, I very much use the 5e rule that "nat20 is just a 20, nat1 is just a 1", but I will flavor an extra outcome or complication. As others have posted, maybe the success took extra/less time, or had a beneficial/unfortunate side effect... the cake not only is well-made, but incredibly artistically detailed (or maybe the cookies are edible, but you forgot the salt!). If there's no real chance for failure though... I forgo the skill check and use "Emphasis Dice" (also stolen from somewhere else, i forget where). Roll 2d20, see which one is farther from 10 and use that as your guideline for the explanation/result. Lower? success with complication. Higher? success with benefit. Something in the middle? normal meh success. (Note that this system inherently has nat20>nat1, as the 20 is 10 "spaces" from 10, while the nat1 is only 9...) And if the dice match, special extra/greater effect just because.
So yes, celebrate the vagaries of extreme dice results, because we are playing a game with dice!
[I played a boardgame at a Con recently called "Xia: Legends of a Drift System". Without going into the game, suffice to say d20s are involved -- and rolling a nat20 gets you one of the 20 Fame Points you need to win! But no more than 1 point per turn this way -- no stacking the situation to roll as many dice as possible just for these points!]
