Why does the case of the "Red-Headed League" interest Holmes?

What interests Holmes about the information given by Jabez Wilson is the strong possibility that Vincent Spaulding is a master criminal known to the detective as John Clay. The most significant details are contained in the following dialogue.

“What is he like, this Vincent Spaulding?”

“Small, stout-built, very quick in his ways, no hair on his face, though he's not short of thirty. Has a white splash of acid upon his forehead.”

Holmes sat up in his chair in considerable excitement. “I thought as much,” said he. “Have you ever observed that his ears are pierced for ear-rings?”

“Yes, sir. He told me that a gipsy had done it for him when he was a lad.”

Holmes has never seen Clay in person but knows of his description, especially that he has a white splash of acid on his forehead and his ears have been pierced for ear-rings. The other features match what Holmes knows about Clay's appearance. He is small, stout-built, and about thirty years old. 

Later when Holmes sees Wilson's assistant on the pretext of asking for directions to the Strand he is sure of his man. He tells Watson:

“Smart fellow, that....He is, in my judgment, the fourth smartest man in London, and for daring I am not sure that he has not a claim to be third. I have known something of him before.”

Obviously Clay would not be working at that obscure little pawnshop unless he was planning to commit a serious crime. Holmes sees the branch bank a short distance away and deduces that Clay is digging a tunnel and wanted to get Wilson out of the way with the Red-Headed League hoax. Holmes deduces that Clay is doing a lot of digging from the moist and wrinkled condition of the knees of his trousers. Later when Holmes, Watson, Mr. Merryweather the banker, and a policeman are waiting to surprise Clay in the bank's basement, Jones the policeman tells Merryweather about Clay's character and history.

“John Clay, the murderer, thief, smasher, and forger. He's a young man, Mr. Merryweather, but he is at the head of his profession, and I would rather have my bracelets on him than on any criminal in London. He's a remarkable man, is young John Clay. His grandfather was a royal duke, and he himself has been to Eton and Oxford. His brain is as cunning as his fingers, and though we meet signs of him at every turn, we never know where to find the man himself. He'll crack a crib in Scotland one week, and be raising money to build an orphanage in Cornwall the next. I've been on his track for years and have never set eyes on him yet.”

Clay is arrested in the act of attempting to steal 30,000 gold Napoleons with a confederate. He is certain to be hanged for murder, but in Victorian times he could have been hanged for his other felonies, including the attempt to steal the French gold. Clay is associated with the infamous Dr. Moriarty, who is Sherlock Holmes' arch-enemy. They will later engage in a death struggle at Reichenbach Falls in Switzerland, which is described in "The Final Problem."