The poems are similar in that both speakers are trying to come to terms with aging and death. However, the speakers differ very much in how they react to old age and dying.
Roethke's speaker is at peace with aging, and describes it in positive terms. He says of growing older:
The lost self changes,
Turning toward the sea,
A sea-shape turning around,—
An old man with his feet before the fire,
In robes of green, in garments of adieu.
An old man before the fire in green robes is a pleasant image that depicts one's waning years as a warm and comfortable time.
In contrast, Yeats's narrator dislikes aging, seeing an old man, in general, unless his soul "sings," as
. . . but a paltry thing,
A tattered coat upon a stick
Roethke's speaker looks forward to returning to nature, to being one with the tom cats and flowers and bees and earth and air. He says "I am renewed by death, thought of my death." Death will bring him closer to the nature that he loves dearly.
Yeats's speaker, on the other hand, feels that nature, the "salmon-falls, the mackerel-crowded seas," is not a "country" for the aged. He dreams of becoming a work of art that will never grow old or die. His image of blissful eternity could hardly be more different from Roethke's. Yeats's speaker longs to be an exquisite mechanical bird:
Of hammered gold and gold enamelling
To keep a drowsy Emperor awake;
Or set upon a golden bough to sing
To lords and ladies of Byzantium
One speaker is at peace with death and looks forward to a return to nature while the other speaker longs to escape death by becoming a beautiful crafted object.