With the announcement at New York Comic Con that Tom Strong and his supporting cast (including his wife, Dhalua, and his daughter, Tesla, plus their friends King Solomon and Pneuman) will be part of DC's upcoming series, The Terrifics, people were understandably curious as to how, exactly, DC owned the rights to these Alan Moore creations from his America's Best Comics line of comics while Moore apparently owns the rights to the League of Extraordinary Gentlemen from that same America's Best Comics line.
The answer to this question goes back to 1998, when Alan Moore was working for Awesome Comics. Awesome Comics was Rob Liefeld's comic book company that he founded after he was pushed out by Image Comics. At Awesome Comics, Moore was working on Supreme with Chris Sprouse and Rick Veitch, Youngblood with Steve Skroce and had plans for a bunch more projects.
However, the late 1990s were a tough period in the comic book industry (Marvel was just getting out of bankruptcy at the time) and Awesome Comics ran into some financial problems and they ended up having to close up shop for a while. Moore actually found out about the news when Wildstorm editor Scott Dunbier called Moore to ask if he would be interested in doing a project at Wildstorm, which was Jim Lee's studio at Image Comics (Image Comics was a collection of individual studios, plus the occasional "Image Central" title). Moore turned him down because he was so busy at Awesome, but Dunbier had heard from another Awesome Comics artist (Brandon Peterson) that the company wasn't doing any more new work at the moment. After Moore confirmed the information, he called Dunbier back and they began to discuss a possible future for Moore at Wildstorm.
Interestingly enough, around the same time, DC Comics had actually looked into buying Awesome Comics to keep them afloat. The problem with that deal is that the best thing that Awesome Comics had to offer was their books that were being done by Alan Moore and Moore was working for Awesome Comics strictly on a freelance basis. Thus, DC could buy Awesome Comics and Moore could just leave, thus negating one of the main reasons to buy Awesome Comics in the first place.
Anyhow, one of the things that Moore was most insistent upon was that he be able to come up with books for all of his Awesome Comics' collaborators, thus keeping them working. So he pitched Wildstorm on America's Best Comics, which were based on doing a new type of superhero comics. Some of the characters were based on the idea of what superhero comics would look like if Superman had never existed, so you had Tom Strong, which was a riff on the old pulp novel hero, Doc Savage.
Of his old Awesome Comics collaborators, Moore created Tom Strong for Chris Sprouse and Greyshirt for Rick Veitch. Top Ten was initially created with Steve Skroce in mind, but instead, it went to Gene Ha and Zander Cannon. Alex Ross recommended J.H. Williams III for Promethea and thus the America's Best Comics line began, as Alan Moore signed a contract with Wildstorm.
You might have noticed what was missing from that above description of America's Best Comics. That's right, there was no mention of League of Extraordinary Gentlemen. That is because League of Extraordinary Gentlemen was actually a separate deal that Moore had worked out with former Wildstorm editor Michael Heisler, before the rest of the America's Best Comics were worked out with Dunbier. That, in a nutshell, is why League of Extraordinary Gentlemen has different rights than the other books.
It is because Moore did the deals separately. The America's Best Comics line (Promethea, Tomorrow Stories, Tom Strong and Top Ten) was a standard work-for-hire situation, while Moore has more rights to the League of Extraordinary Gentlemen. Heisler was no longer a Wildstorm editor by the time that Moore was finishing up his America's Best Comics deal, so Dunbier took over that series, as well, and League was ostensibly released as part of the America's Best Comics line, but it was always under a separate arrangement.
Then, of course, the bottom fell out for Moore when Jim Lee then sold Wildstorm to DC Comics. Since DC had tried to buy Awesome Comics, Moore believed that they were trying to, in effect, acquire him and there is a real possibility that DC was hoping to get Moore's work under their umbrella, but Wildstorm was an attractive asset for DC Comics with or without Moore, so it is unlikely that Moore's presence at Wildstorm was the main reason DC purchased it (Paul Levitz once noted that they assumed that Moore was going to bolt as soon as DC bought Wildstorm).
Moore did not like the situation, but because he had done the deal in the first place to get his Awesome Comics collaborators some steady work, he decided to move forward with the deal. DC promised to let Moore deal only with Wildstorm and Dunbier.
Of course, that turned out not to be the case, as DC got involved a few different times, pulping an issue of League of Extraordinary Gentlemen due to a vintage ad mentioning a Marvel brand of douche and insisting that a story in Tomorrow Stories referencing Scientology be pulled.
After a few years, Moore stopped writing for America's Best Comics and finished out his deal with DC with League of Extraordinary Gentlemen: The Black Dossier. After that, Moore took the League with him to Top Shelf Productions and Knockabout Comics (the latter being the UK publishers of the work. The former handling everything else). He has published the League with them ever since.
Moore's deal allowed Wildstorm/DC to continue to publish the League of Extraordinary Gentlemen stories done while he was at Wildstorm and it also gives them rights regarding the work being adapted into films and television.
The rest of the America's Best Comics are pretty much fully owned by DC Comics, which is why they've had a number of series since Moore stopped writing for them, including a pair of Tom Strong series.
So, that's why DC can use Tom Strong despite DC not being able to use the League of Extraordinary Gentlemen characters.