You have two choices for assigning page layouts; record types and profiles. The combination of options forms a matrix. If you use only one record type, then each profile may be assigned one layout or the other. All records viewed by a user having a certain profile will be displayed according to their assigned layout. If you decide to use record types, then each record type may have a different layout per profile. This potentially allows a user to see different layouts depending on the record they're looking at.
Finally, specific to related lists: if the user can't see the child object, they also can't see the related list. It may be sufficient for you to simply have one page layout, but only allow certain profiles to see the custom object. Users with profiles that can't see the custom object won't see the related list, despite the related list being on the page layout. This offers an additional level of granularity, because you can then assign the View (Custom Object) permission via a Permission Set, giving you user-level granularity to determine who can see the related list, and also simplifies the task of maintaining page layouts, since you now only have one to deal with.