It's sort of a knee-jerk reaction for people to compare the above two series; I thought of Mushishi, too, when I started watching Natsume. There are a lot of superficial similarities: both are episodic, deal with otherworldly beings that are only visible to some, are highly acclaimed, etc. This sometimes leads to bitterness on both sides, as many of these comparisons eventually turn into an argument of which of the two shows is "better." However, to me, these two anime just cannot be compared in this way because both strive to achieve something completely different from the other. Most of the similarities between the two really are just "superficial." Compare the OPs to see the differences in tone: Natsume is light, vibrant, and happy, whereas Mushishi is much more quiet and reflective. More importantly, most of Natsume's themes and even the format of its episodes are completely different from Mushishi--all of which contributes to the differing "overall message" of both.
Though both heavily focus on the interaction between humans and these "otherworldly beings," the nature of those beings determines the nature of the interaction. In Mushishi, the Mushi are, for the most part, unintelligent, primitive creatures that simply exist alongside humans. Mushi are depicted as being a part of nature and are often compared to or even share the appearance of plants and animals. Conversely, the ayakashi in Natsume are anthropomorphic spirits. All ayakashi are capable of conversing and feel the same emotions humans do. Some befriend or even fall in love with humans. Though some ayakashi are barbaric and violent in nature, many are essentially humans in spirit form. In summary, the Mushi of Mushishi are akin to plants, animals, and primordial creatures--all of which are considered lesser than humans, but are accepted as part of nature. The ayakashi of Natsume are supernatural spirits that think and feel as humans do. Though both anime preach that humans should accept Mushi/ayakashi, Mushishi stresses human tolerance of and co-existence with Mushi, whereas Natsume stresses equality and friendship with ayakashi.
These differences are further stressed through the lead characters. Ginko's attitude towards Mushi can be summed as, "They do what they need to do to survive, and if feasible, shouldn't be punished for simply living." The format of each episode of Mushishi is sort of like a medical mystery show, with Ginko playing the role of the doctor: Ginko is first presented with the illness/problem, which he then diagnoses (by IDing the Mushi) and cures (by releasing or relocating the Mushi). Thus, Ginko's role in the show is more of an observer rather than an active participant (with a few exceptions); he helps when necessary, but for the most part, he is the eyes through which the audience views the world of Mushishi. On the other hand, Natsume Yuujinchou is much more of an adventure story; Natsume spends each episode helping an ayakashi (or occasionally a human) with whatever task is given to him. The tasks range from helping an ayakashi be noticed by a human they admire to helping a human escape from a curse that was placed on them by a powerful youkai. Along the way, Natsume often grows to care about his "clients" so much that he throws himself headfirst into life-threatening situations in order to protect them. (Ginko, on the other hand, usually finds himself somehow dragged into life-threatening situations.) And unlike Ginko, who has already come to terms with Mushi by the start of the show, Natsume matures considerably throughout the course of the story and grows to accept ayakashi as an important part of his life by the end. Whereas Ginko is the observer, Natsume is much more of a "hero." (Albeit a considerably weak one.)
The various stories of both Mushishi and Natsume are essentially tales of human nature. Mushishi depicts human nature through humans' actions and reactions towards situations involving the Mushi. (e.g. humans shunning humans who have been afflicted by a Mushi in some way, or conversely, humans accepting those afflicted humans) Natsume Yuujinchou depicts human nature through the protagonist and his relationships with the ayakashi. (As stated previously, the ayakashi are basically humans themselves.) Ginko accumulates knowledge as he travels and encounters different kinds of Mushi and the people affected by them. Natsume's journey is much more personal; even though his childhood was terrible at the fault of both humans and ayakashi, he learns to love both equally and accepts both worlds as important parts of his life. In the end, Mushishi is a collection of stories about humans trying to live in harmonious co-existence with the creatures known as Mushi, whereas Natsume Yuujinchou is about the personal development of the main character, as well as the friendships he forms with various ayakashi.
This is why I get miffed by statements like "Mushishi is a better Natsume Yuujinchou," or conversely, "Natsume Yuujinchou is what Mushishi should've been." The two shows accomplished completely different things; the two are definitely not interchangeable and it greatly undermines the integrity of both series to suggest that one is simply a better version of the other, no matter what your preference is.
That said, I have immense respect for both series, even though Natsume Yuujinchou wins out as my personal favorite. :)
EDIT:
Here is the short-but-sweet AMV that convinced me to FINALLY finish watching Mushishi after putting it on-hold for three-ish years. And wow, this is.... almost publish-worthy, isn't it? I guess I've def. filled up my weekly analysis quota. ^^;;
