In both anime adaptations, the God Hand is first introduced when Griffith uses his Beherit after being rescued by his Band of the Hawk. But in the manga, they initially show up rather early. It is thanks to their introduction in the manga that it can transition to the Golden Age arc of the manga, which covers Guts’ past. In the 1997 anime, Guts first fights the Snake Baron and that does happen in the manga, too. After he fights the Snake apostle in that anime, it instantly transitions to the Golden Age arc without any set up.

In the manga, the beginning extends to another fight with another apostle, the Slug Baron. After the Slug Baron is near death, he uses his Beherit to summon the God Hand and that is when they are first introduced in the manga. In this part of the manga (since it takes about two years after the Golden Age arc), readers are also introduced to Femto, who Guts still refers to as Griffith. Guts attempts to fight Griffith but is easily overwhelmed by Femto/Griffith’s superior telekinetic powers. When Guts manages to escape the God Hand, that is when the manga transitions to the Golden Age arc.

As for the Slug Baron, he’s not directly in the anime but one of his servants, the Hounds Keeper, does play a bigger role in the 2016 anime and is given a more elaborate backstory while he was just merely Dragon Slayer fodder in the manga.

This applies mostly to the 2016 anime series, which serves as a continuation of the trilogy that retells the Golden Age arc. The first season to the 2016 series instantly goes to when Guts fights the Holy Chain Knights. We can assume that the anime did this just so they could get together Guts’ present team in the manga which consists of him, Puck, Casca, Isidro, Farnese (a leader of the Holy Chain Knights), and her half-brother Serpico (with Schierke joining a little later). However, Farnese and Serpico are introduced at the party at midland castle in one of the movies as a fan service cameo.

After the manga returns to the present time after the Golden Age arc, the first story arc is that of the Lost Children, which deals with Rosine, a child who just wanted to be an elf, but she’s really an apostle. This arc is entirely absent in the anime not just so the series can get the team together, but we also have to assume that censors wouldn’t want the hero to kill children (which unfortunately happens in the manga). However, Rosine in her apostle form does have a fan service cameo in both the 1997 anime and the third movie.

Puck, Guts’ elf companion, is non-existent in the original 1997 anime series, but he finally gets his dues in the newer anime one. He is actually introduced as a fan service cameo in the movies riding a wagon, and officially becomes Guts’ companion in the 2016 series. Though the circumstances to how Guts and Puck initially meet are the same, when they meet is entirely different. In both versions, Guts saves Puck from being used as a target for darts practice at a pub (in the manga, Guts flat out kills the darts players while he just beats them up in the anime). However, Guts meets Puck shortly before the fighting the Holy Chain Knights but they meet very early on in the manga. In the manga, they meets shortly before Guts fights the Snake Baron.

In the 2016 anime series, there is an Incarnation Ceremony to bring Griffith back in his human form back to the world of the living. During this part in the manga, Conrad, a member of the God Hand, metaphysically manifests himself through rats to force the people to the ceremony. In the anime, Conrad’s part is entirely removed. In the manga, Guts decides to not fight him since Griffith is not present and wants to save Casca from being burned. However, Slan, another member of the God Hand, does briefly appear during the aftermath in the fight against the trolls (who manifests herself through the slain trolls’ dead bodies) and this happens in both the manga and new anime as she mainly fights Guts with the assistance of the Skull Knight.