Julien Sebag tells me that the constructible topology is useful for the study of the Grothendieck ring of varieties. More precisely, it is relevant to the following question: "if $k$ is a field and $X$ is a $k$-variety with a birational endomorphism $X--\to X$inducing an isomorphism between open subsets $U$ and $V$, are $X\setminus U$ and $X\setminus V$ piecewise isomorphic? You may read about this in this paper.
Another place where the constructible topology is essential is in motivic integration, where constructible sets play the role of the measurable sets of usual integration theory.