Sword of Spirit
Legend
Gygax, in his PHB, explains clerics as inspired by the military orders of the Crusades.
As is shown by my post upthread from St Bernard of Clairvaux - who was one of the authors of the Templar rule - the self-conception of these military orders was as paragons of Christian knighthood.
The paladin is a mechanical interpretation of those ideals of Christain knighthood one finds in the Arthurian and Carolingian romances.
In other words, they are the same archetype: heavy armour; heavy weapons; able to work miracles, including healing with a touch; servants of the divinity, who give expression to the divine providence.
The differences are mechanical, not archetypical. For instance, LoH is a different mechanic from Cure Light Wounds; and casting a Protection from Evil spell is a different mechanic from the paladin's aura of protection; but they are not different in underlying thematic or story terms. Even up to the late 70s, it is quite common to see an evil cleric in an adventure or magazine article described as an "anti-cleric" (just as evil "paladins" are called anti-paladins).
@Matthew L. Martin has (correctly, in my view) noted a tension between the D&D cleric, and D&D polytheism. The AD&D 2nd ed idea of the specialty priest - who does not necessarily turn undead, nor heal with a touch, nor bear the arms and armour of a knight - was one attempt to resolve the issue but - as @LFK has noted upthread - this doesn't really leave much conceptual space for the paladin.
I think this is one of those spaces where ignoring Gary's original intent makes a better game. I mean, even Gary stuck paladins in the first PHB, even though, as you rightly point out, the cleric was designed to be basically the same archetype (sans sword--a not insignificant distinction).