In the Mathias extension (extension by a single Mathias real) the collection $V\cap 2^\omega$ (the reals from the ground model) will have measure zero. This is not the case for the Laver extension.
Both facts can be found in the Bartoszynski-Judah book Set Theory: On the Structure of the Real Line. The former fact is Lemma 7.4.2 while the latter is a consequence of Theorem 7.3.39.