I would like to know the exact differences between Compilations and Album Artists.
I know you can use either one to combine different tracks into a group.
I would like to know how each one behaves relative to iTunes and to the Music app on iOS.
Which one do you use? Or do you use a combination of both?
I read that Compilations is being used less and less in favor of Album Artists.
Is there an advantage of one over the other?
iOS devices don't make use of the Compilations flag. Group by Album Artist is a user switch in Settings > Music which changes which artists are listed, all Artists, or only Album Artists.
Older iPods don't use Album Artist. Enabling at least one of the Compilations entries in the menus makes the artist menu list all artists that have at least one non-compilation track. Disabling both ensures all distinct artists are listed.
In iTunes enter the Songs or a list view, enable the Column Browser. You can then use View > Column Browser > Group Compilations and Use Album Artists to effect similar changes in the iTunes Artists listing.
Take a look at Grouping tracks into albums for gory details.