Yeats describes images of youth (birds in the trees, young in one another's arms) and then an old man as a coat upon a stick. So, the theme sets up as mortality and immortality. That, the world, is "no country for old men." Yeats notes that old men, since they can't be young, can dream of immortality via non-human form (soul). "Out of nature, I shall never take/My bodily form." Therefore, he would take the form of something timeless, as he describes in the final stanza.
Keats, in the second stanza, speaks of the stars looking down on his lover in their eternal gaze, but notes that he would prefer to be mortal and human to enjoy the sensual experience of love; unlike the star which can only observe from afar. Similar theme: stars are immortal; humans are mortal, at least in bodily form.
Contrary to that point, Yeats desires to be like Keats' star. Yeats speaks of being mythical, a figure engraved forever on a golden Grecian sculpture (invoking Keats' Grecian Urn), or Yeats desires to be "a bird" meaning any/every bird serenading all emperors throughout history.
Both poets, in these and other works, speak of immortal concepts like souls and the immortal potential of art itself. But in comparing these two poems, the difference is that Yeats prefers the perspective of the timeless nature (the bird or the golden work of art - or Keats' star) and Keats prefers the human form since that is the full sensual experience of love.