There are some things like those radial tire wires that will penetrate thin or thick tubes alike.
Other things like glass shards seem to barely scratch the surface of the tube, and might take longer to work through a thick tube than a thin tube.
I've started going for the largest tube that is rated for the tire. The reason is that I believe a small stretched tube opens up holes more, and one gets faster leaks. The larger tube would be less likely to open up as much, and one is more likely to get a slow leak, and sometimes one can ride home on the leaky tire/tube.