Does anyone reading this site know of any nuance(s) differentiating these two words? Numerous online sites say that kakejiku is more commonly used, but I have found no information about whether the words are truly synonymous, e.g. with reference to size, significance, materials used, subject of the work, or other characteristics.
At least in Japanese, 掛軸(かけじく) is a type of 掛物(かけもの). 掛軸 only refers to hanging scrolls. Note the kanji 軸 ("axis/shaft"), which refers to scrolls in this context.
掛物 is much less common, but it is broadly used to refer to paintings and calligraphic works that are intended to be hung on a wall. The kanji 物 is just "thing(s)" or "one(s)". They include 掛軸, but they may be drawn also on shikishi, tanzaku, etc.
(English Wikipedia article for kakemono doesn't seem to distinguish kakemono and kakejiku, and they are defined as Japanese hanging scrolls. The Japanese word 掛軸 refers to hanging scrolls in general, and it's perfectly fine to say 中国の掛軸. I don't know how these words are used by English-speaking experts as English loanwords.)