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. That Mathias forcing makes $V\cap 2^\omega$ have measure zero is not unrelated to (as you pointed out) the fact that iterating Mathias forcing increases the splitting number. The Mathias real is an unsplit real: an $x\in[\omega]^\omega$ that for every $y\in V\cap[\omega]^\omega$ either $x\subseteq^*y$ or $x\cap y$ is finite. For a fixed such $x$ the collection of such $y$ has measure zero.
If you're interested in more ways in which the two are very similar you can read Section 3 of Brendle's 'Combinatorial properties of classical forcing notions' which shows that the two forcings have a similar effect on many of the cardinal characteristics in the Cichon diagram.