All is the name. Exfoliated graphite was exfoliated from washed acid-intercalated graphite, the latter being called "expansible graphite". Graphite is first intercalated with sulphuric acid in the presence of an oxidiser (to help the reaction to start), leading to an intercalation compound, which is a graphitic salt. Such compound is washed to remove the excess of acid, and is submitted to a thermal shock for abruptly vaporizing the sulphuric acid located between the carbon layers. Such sudden expansion induces that of the graphite, which is recovered as a very lightweight material: expanded (or exfoliated) graphite. More details in the attached paper.
The difference between various exfoliated graphites and/or acid washed (intercalated) graphites could be in the magnitude (or its distribution) of the lattice parameter c.