They're the same thing.
a "Demon King" is considered to be the single, most powerful demonic entity, often ruling over a large demonic army, while a "Demon Lord" is a powerful demon, usually ranking below the Demon King
The Demon King / Demon Lord is a Stock Japanese Character popular in Standard Japanese Fantasy Settings, where they are usually referred to as the "Maou" (魔王 — literally "Demonic/Sorcerous King" in Japanese) or "Daimaou" (大魔王 — "Great Maou", often translated as "Demon Emperor").
A demon king is a king of demons. Usually in anime, they rally demon armies against humans.
A Demon lord could have different meanings. Some mythologies call high ranking of any monsters and creatures: “Lord” like dragon Lords in slayers anime, which are actually golden dragons who only have more abilities than normal dragons.
I.e. In Arx Fatalis we have Goblin Lords that are basically what in modern anime shown as hob goblins. They are bigger stronger, and higher in ranking. But mysteriously the goblin king in that game, is just a normal goblin, who has turned fat because he eats and loves cookies too much for his own good.
And similarly we have slightly higher ranking demons in Gothic 1 and 2 games who are called Demon Lords. But wait!!! the story is different in anime.
In Anime Demon lord could translate both as Demon king or Demon God. Because lord could also mean God. Sometimes they are called dark lord as well.
Example: Shabranigdo in Slayers anime is regarded both as demon king and Dark Lord. But the 1st description is most likely incorrect.
Why? Because he himself is a creator class being. And he is not just any creatior. But he created not one or two or even three Demon Kings, but 5 of them. And he just created them out of thin air, just like a God (an evil God, however!).
Never underestimate the power of morbid obesity.But mysteriously the goblin king in that game, is just a normal goblin, who has turned fat because he eats and loves cookies too much for his own good.
He was also the eternal adversary of the highest ranking God in Slayers universe: Ceifeed the Flare Dragon God. Now this Ceifeed is also a seriously overpwerde being. He is the creator of 4 Dragon kings that are basically each one Gods in their own rights and are subject of worship by followers of Gods.
Kinda reminds me of Tensura universe where there is 4 true dragons and they are worshipped as god.So Ceifeed is the equivalent of Akatosh in Elder Scrolls lore, or Zeus in Greek Mythology if you prefer it that way. And Shabranigdo is his direct rival and polar opposite.
So in general the Dark lord could mean both a demon King and a higher being such as a Dark God. But I don't think it would be fair to call any demon king a Dark Lord. Like when I hear someone call Muzan or Voldemort: The Dark Lord, it really pisses me off
Like Shabranigdo is someone who could command over an entire universe and even create Demon kings on a whim. And Muzan and Voldemort have problems, just defeating much lower level enemies, and they barely qualify as conntinent level treats. With Voldemort even being lower than Muza.
This is like an insult to the title of Dark lord.