I noticed that Fallout Tactic's version was 1.13. I thought that was a bit silly- why offer a game if it was at its most recent patch? Nevertheless, it's not hard to find the proper patch for it, so that's what I did.
Now I need a CD, apparently. So... what the hell?
Am I supposed to be stuck playing a game that doesn't carry it's most recent patches? Does this mean I need to find my old (scratched) Fallout Tactics disk in order to play now?

The patch you installed probably overwrote the games .exe file and whatever gog does to make it not require the cd with it. Not sure why it would only be patch 1.13 instead of the latest. Your best bet might be to contact gog directly to either inform them that it hasn't got the latest patch or find out why it comes with 1.13 instead of the latest.

According to No Mutants Allowed, the latest official patch is 1.27.
http://www.nma-fallout.com/forum/dload.php?action=category&cat_id=20
The GOG game page states: This game has been patched to the latest official version: 1.27. It would seem that GOG does have the latest version then.
It also shows that there are some other official patches for sound fixes, missing inventory and a bonus mission. I don't know whether these are included in the GOG version or not.