I was previously thinking that the square referred to a public square in the center of town. I do not see how this relates to a public garden. Idiomatic? or just a different way of looking at things. We don't tend to have call parts of downtown public squares in the United States at least in the way I think of cities in Europe having them. City Center in Munich, or inside the Ringstrasse in Vienna, etc. Areas where there is a plaza (generally with a cathedral). I don't associate that with a garden though so maybe I am completely off track.
un carré = a geometrical square, the second power, or a patch
une place = a plaza or square as in a public square; un square seems to be a synonym
un jardin public = a public garden, which may or may not be une place
une case = a square on a grid
Honestly, words like these are a lot easier to learn with a proper context.
"Jardin" means a garden in English. "Un jardin public" means a "park". Keep in mind that in certain parts of the world, the word park is synonymous with square. So, if we were in Egypt for example, I would be telling you about the Square of el-Tahrir', while in fact I mean "the park of el-Tahrir." That's British English.
From all the comments it seems to me that there are different approaches to what constitutes a "square" in France, in the US, in Britain c. I just want to ask any Francophone: would you call this https://commons.wikimedia.org/wiki/File:%C4%8Cesk%C3%A9_Bud%C4%9Bjovice_Square.jpg "un jardin public", given the conspicuos lack of plants, or could this ONLY be called "une place/un square"?