This subjective question requires a subjective answer. What this means is that the reader determines the strengths and weaknesses of the story on a personal level.
I find the story "The Devil and Tom Walker" to be a wonderful piece of American literature. Therefore, the weaknesses I would find are nonexistent.
As for the strengths, I find that the descriptions of the setting and characters are immaculate. The fact that the Devil is described in many different ways adds to his character:
"And pray, who are you, if I may be so bold?" said Tom. "Oh, I go by various names. I am the Wild Huntsman in some countries; the Black Miner in others. In this neighbourhood I am known by the name of the Black Woodsman. I am he to whom the red men devoted this spot, and now and then roasted a white man by way of sweet smelling sacrifice. Since the red men have been exterminated by you white savages, I amuse myself by presiding at the persecutions of quakers and anabaptists; I am the great patron and prompter of slave dealers, and the grand master of the Salem witches."
"The upshot of all which is, that, if I mistake not," said Tom, sturdily, "you are he commonly called Old Scratch."
"The same at your service!" replied the black man, with a half civil nod.
Here, Irving is showing how is is trying to relate to the fact that many different people have different descriptions of the Devil. Through this acknowledgement, Irving is able to bring in, and engage, more a more deepened reading.
As for the story itself, the story allows for the reader to assume what happened in many places (for example, what actually happened to Tom's wife?).
For me, given the subjective nature of the question, I find no weaknesses to exist in the story. For me, the story only supports Irving's power as a writer and storyteller.