I can throw in my personal experience here.
It is, without doubt, much easier to learn one and then the other. I learned Japanese before I learned Chinese, having studied for about 2 years before properly taking up Chinese (before that point I toyed with the idea and did some research, but never anything serious or long-term). The areas it's has helped are, to me:
1) Writing Hanzi
Japanase uses "kanji" (漢字) for roots of nouns and verbs, which it imported from China around 1500 years ago. You say kanji are "derived" from Chinese characters. Most of them are actually exactly the same. A small number follow simplifications nearly the same as on the mainland, and a very small number have simplifications unique to Japan.
By learning Japanese, I already knew about 1000 hanzi when i switched to Chinese. The simplifications were easy to get my head around because they conformed a lot to simple rules (with a few exceptions), and I easily also picked up traditional characters, as Japanese deviates from 繁字体 a lot less than the mainland.
My main complaint from other people about Chinese is hanzi (and grammar close behind). I already started Chinese with a big character library, and I've never had a problem with them. A lot of beginner Chinese vocabulary is just common sense, and this very much makes at least beginner level learning very easy to adapt to.
But, it's worth noting that the stroke order is pretty different between the two languages, and I've often been commented on by my Japanese teacher that I have bad stroke order when I'm just using the Chinese stroke order (and even then, the mainland has a different stroke order to Taiwan).
Around 60% of Japanese words are 漢語, which is about 18% of speech. This is comparable to how knowing Latin makes it easier to learn most European languages, all of which take some 20% of their words (usually more) from Latin roots.
Learning Japanese doesn't teach you a lot of fundamental vocabulary, but it's usually not impossible to figure out basic Chinese words' meanings. The technical language relates much more closely, in words such as 特別行政区、日本、and even 4 character proverbs such as 一日千秋. A lot of modern technical Japanese is more Anglicized, so not as easy to follow Chinese roots, however, but a lot of the slightly older or scientific terminology is very alike, and relatively easy to make the switch.
Also, you come to learn what characters mean by knowing Japanese equivalents. As mentioned already, verbs like 走る mean "run", contrasted with 走 in chinese.
3) Pronunciation (All dialects)
In these loanwords from Japanese, they also take (primarily) middle Chinese pronunciations of the characters. 漢字 "kanji" from "hanzi", 我慢 "gaman" from "ngaman" and so on. Japanese is much more complicated because it loans so much more from different areas and regions of China, so converting to exclusively one dialect is much easier (and there are far fewer pronunciations per character).
Another thing to note is that these pronunciations tend to be from middle chinese, seen as an ancestor to most dialects. Mandarin nowadays is a totally different language, but still resembles the sounds they once had, and the rules make it easy to guess how a chinese character sounds in any dialect, knowing even just one Japanese pronunciation. It doesn't help so much with tones though, but nothing really prepares you for those unless you know another Chinese dialect.
Examples include: /k/ in Japanese is very often /h/, or /j/ in words that used to be /k/ in Mandarin (like Peking -> Beijing), long vowels usually reflect -ng finals in Chinese. I have a longer post about these on Japanese Stack Exchange
4) Languages get easier the more you know
The more languages you know, you tend to pick them up a lot faster. You get familiar with the type of grammatical structures any language can have, and just get your head around thinking in a different tongue to your native one. After learning Japanese and Chinese, Korean has quite simply been the easiest language I've ever learned, due to the ~60% vocabulary similarity, and near 90% grammar similarity to Japanese and Chinese. Even when I tried my hand at Russian, that was way easier too.
It's not just me; most people who are at least trilingual have similar experiences. I don't think I need to cite this one so much.
A long list, but pretty much why I think it's easier to learn one then the other. Because of the direction of loanwords though, I think Japanese to Chinese is much easier than Chinese to Japanese. I also have found a similar in switching to Korean that it's much easier after knowing other Asian languages.
 A study from 国立国語研究所 (National Institute for Japanese Language and Linguistics)