First off, in Steam on the machine which isn't syncing, right-click Stardew Valley from the Steam game list on the left and select Properties. In General, make sure Keep games saves in the Steam Cloud for Stardew Valley is checked.
If it's checked already, let's try to convince it to really get those saves from the cloud: uncheck the Keep games saves..., close the Window, and Exit Steam. Then start Steam again, go back in there and check it again. Maybe wait a few seconds/a minute, then try playing Stardew Valley again and see if it syncs the saves.
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If you're playing on a PC, your local saves are located here:
C:\Users\*yourusername*\AppData\Roaming\StardewValley\Saves
This is where Steam should be syncing each save folder to from its Steam cloud location. As an example, my current active save in there is in the folder BeachyKeen_268602465. If the above procedure didn't work and you still want to want to play the same save on that second machine, you'll need to copy your active Stardew Valley save folder (should have the most recent save date on that folder) from the working machine (in my case, the BeachyKeen_268602465 folder) and put it in the same place on the non-working machine. But then you need to keep track of which machine has your most recent save as you have become the cloud save.
I do this all the time as I have more than one six computers I play SDV on and I don't use Steam (I use GOG but not its cloud save Galaxy client). So I move my save around manually but this is not a good long-term plan unless you're either very disciplined or nuts.
Or both, like me.