Mushishi is perhaps the greatest storytelling I've seen from any anime. This is not for some epic overarching plot, but rather for a minimalist episodic approach that gives you new characters, new settings, new stories, and new mushi to deal with each episode. Other episodic anime may have the tendency to lose your interest partway through, but Mushishi is special in that each episode will leave you totally and completely satisfied.
After an episode of Mushishi ends, there is no need to rush in to the next one to conclude the cliffhanger from the previous episode. That's because there is never a lack of a conclusion in each episode. Each episode is its own story, completely unique from all the other episodes. When I watched Mushishi, I'd see an episode, be satisfied, and when I was in the mood I'd watch another. I was never addicted, but it never fell off my radar. It held my interest consistently throughout the two month period that I watched it, and while some may praise shows that make you want to watch it all in a day, I view that as a weakness. It seems to suggest that you were never satisfied with a single episode, and that to get your "fix" you had to burn through many at a time. That difference is the biggest strength to Mushishi, and what gives it its lasting appeal.
Another strength of the show is its beautiful rendering of nature. Of course, when the show is steeped so heavily in nature you would expect this, but there are some truly breathtaking scenes in this anime. The only flaw in the animation is the lazy character design. Humans from episode to episode are indistinguishable from each other, and Ginko seems to only be recognizable through his white hair and green eyes. This feeds into the other big flaw in the show: the characters. No character other than Ginko is ever fully realized. We can feel for some of their problems and emotions, but what we are really left with at the end of the episode is the story they were used to tell. This is fine: it's an episodic anime and no one would expect these characters to be terrificly developed, but it may be an issue for some people.
Finally, I must talk about how each episode fades into a new ending piece. It gives the viewer a sense of closure with each story: that what they had just viewed was not just a piece of something larger, but rather a work of its own.