What is important about the title "The Devil and Tom Walker"?
Washington Irving, when selecting a title for his short story about a somewhat pathetic, downtrodden and unhappily-married man named Tom Walker and this character's encounter with the devil, The Devil and Tom Walker, was probably emphasizing the centrality to his narrative of the enduring relationship between the two titular figures. While Old Scratch, as Tom addresses this mysterious figure he encounters in the woods, figures only briefly in the beginning and end of his story, Irving almost certainly wanted to ensure that the ill-considered arrangement to which Tom agrees during that encounter remains in the reader's subconscious.
Tom Walker's deal with Satan, as is usually the case when fictional characters seek professional and financial advancement via unholy arrangements, involves his agreement to turn over his soul to the devil in exchange for that advancement in "this" life. Whether Tom is fully cognizant of the complete details of his agreement with Old Scratch is not really certain. After all, Irving rushes through the arduous process of negotiating in deference to brevity and, ultimately, surprise. The following passage from The Devil and Tom Walker constitutes the totality of this process:
"The black man told him of great sums of money which had been buried by Kidd the pirate, under the oak trees on the high ridge not far from the morass. All these were under his command and protected by his power, so that none could find them but such as propitiated his favour. These he offered to place within Tom Walker's reach, having conceived an especial kindness for him: but they were to be had only on certain conditions. What these conditions were, may easily be surmised, though Tom never disclosed them publicly."
As noted, actual appearances of the devil in Irving's story are fleeting. They are, however, absolutely central to the narrative. Much of the story details Tom's exploitation of those in need of financial assistance with an emphasis on his extreme greed and narcissism. The story ends with a very brief and very sudden return of Old Scratch. The devil's presence, however, is felt throughout the story, and the author's decision to title his story The Devil and Tom Walker served to emphasize the former's inordinate if unseen role in the latter's life. The devil may have been unseen, but he was always by Tom's side.
The title "The Devil and Tom Walker" might have been chosen to make it very plain that Tom Walker was dealing with the devil, and not a lesser evil spirit, or a human dressed up as one. However, "Old Scratch" is repeatedly referred to as "the devil" throughout the text, so it seems that this interpretation provides no new information that is not already explicit.
However, I think one nuance here is that the title refers to "The Devil", with a capital D. While this may simply be a matter of grammatical accuracy, capitalizing the D in devil usually implies that one is referring to Satan/Lucifer, the Prince of Hell, etc. "The devil", as he is referred to in lower-case throughout the text, typically means ANY devil, not "the Devil Himself". Thus the title may serve to tell us that it was indeed Satan himself that is involved.