Why is odor spelled odour, and of what significance is the spelling, in the short story "Odour of Chrysanthemums?"
That "our" spelling to the modern eye generally demarcates a difference between American English and the English spellings found in the UK, Canada, Australia, and most other parts of the English-speaking world. It doesn't indicate any difference in the pronunciation of the word, nor does Lawrence use it for any conscious reason in this story except because it was the way he was taught to spell the word "odour."
As to why the spelling differs, the "ur" ending originally reflected the fact that these words entered English for the most part through French, rather than directly from the Latin. However, etymological confusion led the "u" to also be retrospectively added to words which were direct Latin borrowings. Webster's dictionary is generally given the credit for the fact that the "or" endings for these words are the standard in modern American English; most early writings from the USA use the "our" spellings, which were overwhelmingly more common in English at that time. Webster deliberately chose the "or" spelling as a means of returning English to its Latin roots and standardizing what he saw as an unnecessarily complex system. Webster is responsible for a lot of the aesthetic spelling differences between American and British/Canadian/Australian English today.
The answer is simple enough: "odour," with that extra "u," is the British spelling of the word "odor," which is the American spelling. It is not particularly unusual for an English word to have different spellings in the UK and the US, as that's just the way the language developed over time. There is no special significance to the added "u" in odour beyond this cultural difference, though the spelling does clue the reader in to the fact that they are reading a story that is by a British writer and set in the UK, right from the very title.
The story was first published by The English Review, a UK-based literary magazine, so the spelling can also be said to reflect the editorial standards of that particular publication.
The only difference is that "odor" is the American spelling and "odour" is the British spelling for the same word and meaning. D.H. Lawrence in a British writer, which is why the word is spelled with the "u."