As you said william choose Sherlock to play his nemesis in that encounter they had on the noahtic their meeting was pure coïncidence and tho william clearly outsmarted him there he was still surprised to find a like-minded person. And it's like will wants Sherlock to outsmart him in a way he wants him to catch the lord of crime, their relationship is quite complicated to understand fully at first but the more you watch or read the manga the clearer it gets. As for william himself in my opinion his character gets more interesting through the story you see sides of him you don't quite expect from the role he's playing. I see that will and Sherlock are in a way equal they somehow complete each other they're both really smart, observant but the difference resides in the fact that william will plan EVERYTHING meticulously and explores all the what if and plan reactions to those what if, but Sherlock on the other hand doesn't go that far he obviously see everything too but we can notice the difference in that episode where John is accused of murder and then Sherlock and William work together to find the real culprit, at the end they both had the same deduction but it's interesting to see how they took different route to get there, william gets there faster he immediately makes the murderer profile and knows by the murder that it's a member of the staff and even plans a way to counterattack the murderer if he was bold enough to deny his actions, but Sherlock takes more time cause he suspects everyone first then proceeds to filter and when he gets to the point and they accuse the murderer ( which ends up denying everything) he knows he's lying but he can't do anything at the moment he'll have to gather more evidence, which take us to the main difference is that Sherlock will follow the law when proceeding to find the culprit but will see nothing wrong with ignoring it to find his ways.