I noticed that when I originally bought the game the setup file that I downloaded and installed was named: setup_ftl_advanced_edition_2.3.0.13.exe

But now, if I go to re-download it, it is called: setup_ftl_-_advanced_edition_1.5.13_(9345).exe

The files are different sizes, but both seem to install version 1.5.13 of the game.

What is the difference?

You are right, there are no differences for the game itself.
My guess would be that GOG is simply rolling out new installers, which change the way things work internally a bit? It could be support for Galaxy, for example.

From OP it sounds like they're now actually naming installers in a way that we can tell the version of the game from the installer, instead of GOG installers having their own freaky version numbering that's totally different.

About time I say! Should make it much easier to see if we're up-to-date here at GOG compared to others...

Please see the changelog. Recent windows version had only one thing change - galaxy support. Probably different name of binary and different installer structure.