I'm probably going to get grief for this, but IMO the reason the paladin might feel like the poor cousin to the cleric is because the cleric has been buffed to heaven (pardon the phrase).
In earlier editions of the game, the paladin was a fighter subclass that got some access to clerical abilities before multiclassing was really a thing. It gave you a little bit of turning, smiting, and healing on your fighter in exchange for having to live up to a moral code. The cleric, by contrast, had turning, smiting, and healing, but was very restricted in weapons (impact - or, at the time, bludgeoning only) since the clerical code was "couldn't draw blood" with weapons. In earlier editions, swords were the weapon of choice as they could be extremely magical, have intelligence, special abilities, etc. - and the paladin of course could take full advantage of the +5 Holy Avenger whilst the cleric could not.
I believe it was 2nd edition AD&D that started introducing "specialty priests" that could run around with all the same weapons as fighters (especially swords). We've also seen a lot of abilities go to more and more classes (for example, things like "martial weapon proficiency" have become common). Clerical spells at low levels were also mostly defensive/buff (at high levels, they did get powerful offensive tools like Flame Strike and Holy Word), but as time has gone on, they have stolen more and more offensive spells from the arcane list.
The reason the paladin feels like a poor cousin of the cleric is that clerics have been steadily agglomerating everything that every other class does and have now agglomerated enough fighting toys that they're more or less on par with the paladin.
And this is probably going to be an unpopular sentiment (and probably come across as grognard) but I really think class design benefits a LOT from saying, "no, you absolutely can never do X" - while I know players have chafed at restrictions from the beginning (witness early articles about anti-paladins, and other off-LG paladins in Dragon magazine and the recent cheering of moving away from alignments altogether - this is a whole other subject), to me, the paladin was always defined by what it COULD NOT do (anything not lawful good) as much as by what it could do. But for game/class design in general, carving out a niche for a class to fill usually means by definition you have to give it "your class can do X" but also make sure you pair that with every other class being told "your class CANNOT do X."
You know, the whole, "when everyone is special, nobody is." When a character has access to everything, they no longer need an adventuring party. The whole idea of giving everyone something they CAN'T do is to allow someone else that CAN do that thing to have a turn in the spotlight. Players love multi-classing for this reason, and I think multi-classing in its current form is a big part of the problem here with being able to accumulate lots of diverse abilities.
So to get this back on topic, if the question is whether or not the paladin is obsolete because of the cleric, the problem is not that the paladin has gotten worse (though I would argue the move away from alignment - and thus away from requiring LG - has taken away some of what made the class special to me, because it required a player to offset mechanical advantages with roleplaying decisions) but rather that clerics have gotten too much better.
Apologies to whomever mentioned it upthread, but I think "a paladin is a servant of alignment/virtue" while "a cleric is a servant of a particular deity" is probably the right answer on how you differentiate them. And because a paladin's devotion to ideals is more pure than a cleric's (whose ideals have to be mixed with the dogma of a particular deity), the paladin class should be able to be more focused in what it does than the cleric class. Of course, multi-classing in the way 3e introduced it also did a lot to kill the specialness of the paladin because now you literally CAN be a fighter with a smattering of clerical abilities or a warpriest with a smattering of fighting skills.