There is no difference between an overcoat and a coat. An overcoat is an archaic version of a coat. Coats are full length, to the knee or longer and can be worn over a jacket, which is why they were called overcoats, this has now been abbreviated to coat in modern usage
Actually, the fireplace ledge is usually spelled mantel, or also mantelpiece. A mantle is an overcoat, but a very specific type of one, often without armholes, rather like a cloak. It's seldom used anymore, probably because people rarely wear them anymore. :-)
Interestingly, a Google Image search on "mantle" throws up more images of the earth's mantle (an outer layer of the earth's core) than anything else.
I quickly went through all the images that Google Graphics showed me for "jacket" and I didn't notice anything that I'd call "płaszcz". I also googled "parka" (which sounds Polish to me and apparently is a word in Polish as well), and most results look more like a "kurtka" to me and are called "kurtka typu parka" for example, although "płaszcz" has been used with some of them, true. But I don't know if it's enough to change the accepted translations here, we'd like to show the difference somehow...