No matter how many times I change the climate, press save and then exit the creation kit, the changes never show up in game. What gives?

Are you testing with a clean save?

I didn't. How do I get it to work on my current save?

The Edits may not show up in your current save as some things need to load in a new game. You can use the console from the main menu to "coc" (center on cell) to enter Tamriel as the default character almost as if starting a new game without having to go through the entire start sequence. Just type coc whiterun and you will be placed on the bridge over the White River just outside of Whiterun. You may then notice your edits.(Don't Save)

For future reference, when testing a mod this is what I do. Deactivate your mod or remove the esp from the data folder temporarily. Go in game and Create a save and give it a unique name like MyModInstalled. Then go to your mod manager and activate your mod. Then turn off the automatic save feature in the settings menu and also save on wait and save on fast travel. Use that save to check out your mod, but DO NOT save on exiting the game. If you need to do more tweaking, delete that save you created and make a new one. That way you will be testing on a save that has never seen your mod.

"Clean save" doesn't have to be a new game. It simple has to be a save that was saved WITHOUT your mod active. This ensures it's freshly loading in the data, and eliminates the chance of stale data causing issues.

If you saved with your mod active, then make changes and load that save, you may not get accurate results.

I keep a save with just Skyrim when I get out of Helgen. Then I add the DLCs and make another save. I use those saves to test mods. I'd like to get a save with just the DLCs and all major quest lines completed to test certain things but I never seem to get that far with no mods.