How is the aromaticity in graphene different from the aromaticity in benzene?
Simpler explanations, can by their nature be less accurate than a detailed description, but are helpful in beginning to understand things.  Getting to the heart of your question:
“How is the aromaticity in graphene different from the aromaticity in benzene? . . .  but is it possible to find a simpler if less accurate way to explain the difference,”
Perhaps a productive way to approach your question [portion given above] is to consider the molecular structure of coronene and circumcoronene while contrasting them to the interpretation of aromaticity of graphene given in your cited article.
 circumcoronene
 CORONENE
Graphene from article cited in question.
According to the computational method employed “adaptive natural density partitioning method” [we can choose to agree or not agree with their interpretation as indicated by a comment to your post by @A.K.], the aromaticity of graphene is centralized within the set of hexagonal rings.  To quote the authors “Aromaticity in graphene is local with two π-electrons delocalized over every hexagon ring.”  The salient point here being the “delocalization” is within and isolated to or restricted to [my words] a six-membered ring.
As discrete molecules, the aromaticity in coronene and circumcoronene, as depicted the PubChem structures, is delocalized throughout the entire discrete molecule and not restricted to a discrete set of hexagonally bonded carbon atoms.
In other words, it comes down to electron density distribution.  In graphene, according to the authors of the article cited, the areas of highest electrons density are within the center of the six-membered ring while in coronene and circumcoronene the electron density is more uniformly distributed over the entire molecule.
Ref:
https://pubchem.ncbi.nlm.nih.gov/compound/coronene
https://pubchem.ncbi.nlm.nih.gov/compound/Circumcoronene