Hesiod, Theogony 758 ff (trans. Evelyn-White) (Greek epic C8th or C7th B.C.) :
“Nyx (Night) carries Hypnos in her arms, and he is Thanatos’ (Death’s) brother . . . And there [near the house of Nyx in the underworld] the children of gloomy Nyx have their houses. These are Hypnos (Sleep) and Thanatos (Death), dread divinities. Never upon them does Helios, the shining sun, cast the light of his eye-beams, neither when he goes up the sky nor comes down from it. One of these [Hypnos], across the earth and the wide sea-ridges, goes his way quietly back and forth, and is kind to mortals, but the heart of the other one [Thanatos] is iron, and brazen feelings without pity are inside his breast.”
Where Haides is the Lord of the Death, Thanatos is the God or Daimon of Death itself. Though the occasional sacrifice was offered to Thanatos in specific, He was given no temples or public altars and shrines.
The differences between Theoi and Daimons in ancient Hellas seems a bit blurrier than some people believe, and one region’s Theos would be another’s Daimon, or even one person’s Theos would be…, and so on. This is why I don’t whinge about how the word “polytheism” somehow implies a lack of acknowledgement and/or honour given to spirits (and no, this is not a strawman, I’ve seen a few people claim this), and why, most importantly, I don’t describe my beliefs as “both polytheistic and animistic”, because traditional polytheism tends to imply a degree of belief in peripheral daimones. In Hesiod’s Theogony, the birth of Thanatos seems to imply a Daimonhood, but I don’t really have a concrete opinion of one way or the other in regards to Thanatos.
Basically, Thanatos’ existence as a separate entity from Haides is why I specifically describe Haides as “Lord of the Dead”, and ancient thought does seem to have a similar separation of Haides from being “God of Death” and instead describes Him as “God of the Underworld” or “God of the Dead”. Where Haides governs the dead, Thanatos delivers death. Thanatos also, even per Hesiod, seems to exist outside of Haides’ governance, and acts outside of Haides’ order.