Intuitively, closure of SL_{2} under intersection is reasonably easy to see, particularly if one considers the Myhill graphs of the automata. Any path through both graphs will be a path through the intersection of the graphs (by which we mean the graph resulting by taking the intersection of the vertex sets and the intersection of the edge sets).
For the union, on the other hand, the corresponding construction won't work. An automaton built from the union of the two automata will still recognize all of the strings in L_{1} and all of the strings in L_{2}, but it is likely to also recognize strings made up of adjacent pairs from L_{1} combined with adjacent pairs from L_{2} that aren't in either language. And, in fact, we can use Suffx Substitution Closure to show that there are languages that are the union of two SL2 languages that are not, themselves, SL_{2}.