Difference between Laver's and Mathias's forcing

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.

Another interesting difference is in the available subforcings. Like a Sacks real, a Laver real $r$ is an object of minimal degree; whenever $x\in V[r]$ if $x$ doesn't belong to the ground model then we can recover $r$ from $x$ (ie $V[r]=V[x]$.) This is a special case of Theorem 7 in Groszek's 'Combinatorics on ideals and forcing with trees'. On the other hand this fails badly for Mathias forcing. If $m\in[\omega]^\omega$ is a Mathias real, and $m_0,m_1$ are subsets of $m$ with $m_0\cap m_1$ finite, then $V[m_0]$ and $V[m_1]$ are distinct extensions. (This is Corollary 8.3 in Mathias's original paper 'Happy Families').

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.

In this paper Alan Dow compares Laver and Mathias forcing regarding their effects on the algebra $\mathcal{P}(\omega)/\mathit{fin}$. He considers four iterations of length $\omega_2$: iterate $L$, iterate $M$, iterate $L*M$ or iterate $M*L$ and calculates the values of many familiar cardinal characteristics of the continuum in the resulting models.