I have noticed that anime adapted from a manga heavily tone down the level of violence and "fanservice" (sometimes skipping them entirely). Why is the censorship level different between manga and anime? I assume they are bound by the same censorship laws, or is this not true?
There's a Manga Censorship panel at Comic con that talks a bit about habits in Manga, of note:
Fujimoto also pointed out the particular demographic stratification of manga into shonen, shojo, seinen and josei (boys, girls, men and women) genres, resulting in a product range that spans all ages. Each of the various manga magazines published in Japan is targeted to one of these specific demographics, resulting in about a 50-50 split between manga aimed at those under 18 and manga aimed at adults.
Additionally, where money made from manga is primarily sales as opposed to something like an anime TV series where there's both a considerable investment from the studio, sponsors, and in most cases licensing for products (like figures, T-shirts, etc.), the panel also noted that:
Kanemitsu opened by saying that, despite the most idealistic wishes of progressive thinkers, Japanese publishers may not necessarily be fighting at full force against censorship. Other disconnects also exist between the definitions of obscenity and child pornography in Japan versus the United States, and also how the different cultures conduct business.
In anime, sponsorship and how much a TV studio has to play in anime productions is a much greater influence than the stores that sell manga has over mangaka and editors. Cowboy Bebop was originally taken off the air mid-way when it first aired because TV Tokyo said it was too violent. There were purportedly studio and sponsorship pressure on GAINAX during the original run of Evangelion.
Then you have late-night and satellite TV anime series which show nudity and violence, or OVAs which, as long as properly labeled, can be borderline pornography. The former has lax studio pressure and the sponsors already know their demographic. The later is not much different from how manga is sold and distributed, at a store where titles that would normally be censored are labeled appropriately or are in their own sections.
As quoted above, because of the way manga is distributed, there's a 50/50 split between manga produced for under 18 and those for adults. So you have a lot more content that's risque or violent in manga because of the split. A few decades ago when most of the distribution of anime were straight to video, a lot more shows contained nudity and violence, even ones that weren't necessarily aimed at adults. But because of the distribution method, the animation studios themselves had a lot more control over what they could write and animate. Around the mid-to-late 90's, TV studios started airing more and more shows, and they had their own guidelines on what they would and wouldn't air so even the incidental nudity and blood were taken completely out if animation studios wanted to produce a TV show that aired on a national station during the day. Note that when a show that aired on TV, that needed to be censored so the stations would air them, makes it onto physical media like DVD's and Bluray's, those white/black washes, glares, steam, etc. are all removed and are uncensored.