A good example is testing if a graph is connected.  This is easily seen to be $\mathsf{MSO}_2$ expressible, since a graph $G$ is connected if and only if for all partitions $A \cup B$ of $V(G)$, there is an edge between $A$ and $B$.  On the other hand, it is not possible to test if a graph is connected in $\mathsf{MSO}_1$ (or even in existential $\mathsf{MSO}_2$).  See this [paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.72.2575&rep=rep1&type=pdf) of Fagin, Stockmeyer and Vardi for a short proof via Enrenfeucht–Fraïssé games, which is easier than an earlier proof due to Fagin.