Nah, I'd say it's more about the Japanese getting fed up with their unreadable runes and deciding to switch to vaguely more readable ones. 蜘蛛 is kanji and also reads kumo, I believe (At least I'm fairly sure it's the "gumo" part of Tsuchigumo. ) But it also looks like two blocks of black with white dots to me.Wait, the spider "kumo" is written in katakana meaning there's probably a language that spiders are called "kumo" or something like that. So what language is that from...if anyone knows.
Good riddance, to be honest. Wikipedia also says modern Chinese loanwords are also now written in katakana, and typing 蜘蛛 redirects you to クモ in Japanese Wikipedia. Either that or Japanese Wikipedia is simplifying things for convenience.
Edit: Yes, it's Wikipedia simplifying things for convenience. Disregard the katakana version, kanji seems to be the correct usage. Why doesn't it give a disambiguition page of cloud/spider is beyond me. I don't even know Japanese anyways, so it's none of my concern.