Burnt Norton's resemblance is not so superficial. The place from which the poem takes its name is something like Clarissa's Bourton, a manor house he once visited having an ethereal rose garden. Possessing uncannily similar names, both are in Gloucestershire and both are freighted with nostalgia of nearly Edenic intensity.
Both the poem and the novel have a circular structure, returning at end to the point of departure, though this is less obvious in the novel. Both use the imagery of time to convey esoteric meaning: the flow of time; the experience of past, present and future; the illumination that accompanies a release from the confinement imposed by time. For both the word that best describes the enlightened state is love, not of physical desire, but a metaphysical acceptance of all and everything -- good and bad, important and trivial, handsome and ugly, permanent and transient. Both evoke nonexistence as counterpoint to the physical world, what Eliot calls the still point in the gap between un-being and being.
The big difference is one of religious faith. Eliot's point of view is entirely Christian, his faith in redemption and eternal life. Woolf neither accepts nor denies religious faith. It, like everything else, is part of life, but is not life itself. Woolf's point of view is more complex. She gives us the minute details of each living moment to treasure. She gives us our lives to live, fully if we can, but without any certainties, lacking all permanence, and having no freedom from doubt. Woolf's imagery deals with unfathomable nature -- that of oceans, skies, and piercing light -- and also much natural beauty -- of trees, gardens, and bounteous flowers. Her essential theme, I believe, is the peculiar courage that can face up to aboriginal fear -- fear of non-being, fear of death -- and, having allowed ourselves to exerience this deep amorphous dread, do more than survive, ourselves shine forth in fully-realized humanity. This is very different from Eliot's vision of purification, perfection, and spiritual unity.
Early in Mrs. Dalloway, Clarissa decides she should mend her dress. Woolf uses the dress itself, her acquisition of it, the maker, and its need for repair to show us Clarissa's compassion, ability to live life as it is, and potential to experience fear -- existential dread I would say -- as a necessary part of life. Using fairly complex imagery it's possible also that in this passage Woolf intimates Clarissa's capacity for enlightenment, the epiphany Clarissa will experience at the end of the novel which will reveal her fully to us as existential heroine, transformed but still present ("there she was") and still the person who will do for herself that she was at the outset.
Here is the passage:
Quiet descended on her, calm, content, as her needle, drawing the silk smoothly to its gentle pause, collected the green folds together and attached them, very lightly, to the belt. So on a summer’s day waves collect, overbalance, and fall; collect and fall; and the whole world seems to be saying “that is all” more and more ponderously, until even the heart in the body which lies in the sun on the beach says too, That is all. Fear no more, says the heart. Fear no more, says the heart, committing its burden to some sea, which sighs collectively for all sorrows, and renews, begins, collects, lets fall. And the body alone listens to the passing bee; the wave breaking; the dog barking, far away barking and barking.
Here's Emerson's poem. It's one of only a handful of really good ones that he wrote during his long lifetime and it first appeared in the very first issue of The Atlantic Monthly in November 1857.
by Ralph Waldo Emerson
If the red slayer think he slays,
Or if the slain think he is slain,
They know not well the subtle ways
I keep, and pass, and turn again.
Far or forgot to me is near,
Shadow and sunlight are the same,
The vanished gods to me appear,
And one to me are shame and fame.
They reckon ill who leave me out;
When me they fly, I am the wings;
I am the doubter and the doubt,
And I the hymn the Brahmin sings.
The strong gods pine for my abode,
And pine in vain the sacred Seven;
But thou, meek lover of the good!
Find me, and turn thy back on heaven.
Eliot's poem is widely available. An internet search turns up a number of locations where you can read Burnt Norton including this one.
He is the much greater poet; they are not in the same league at all. And Four Quartets, of which Burnt Norton is a part, is a masterpiece.
There's no intersection between the lives of Woolf and Emerson, of course, and no reason to expect that Woolf had more than passing interest in his poetry. Eliot, on the other hand, was more than an acquaintance of hers. She and Leonard published his poetry -- hand-setting it for printing at the Hogarth Press which they jointly ran. And Eliot is mentioned in her diaries, though it's not evident that his work had any influence on hers. Some anecdotes of Eliot and the Woolfs, along with a description of their early publishing business, is given here.