Relationship between the Shin Sekai Yori manga and anime? I have seen the anime series and I just discovered that there is a manga.

It seems to be publishing at this moment, and I could not determine where the manga fits in.

WIthout spoiling the actual manga story, can someone elaborate if this is a different story or not?
SeventhStyle's review (quoted in Dimitri mx's answer) touches on the major issue with the Shinsekai Yori manga. The manga has brazenly explicit visuals, unsurprisingly almost entirely of Maria and Saki3 engaging in lesbian antics.1 Sure, (homo)sexuality was a major theme of the novel, but the manga really is just in bad taste, going out of its way to have Maria and Saki fornicate whenever they get a free moment. It's pretty clear that the manga isn't targeted at the same mature demographic as the novel was.2
In any case, the plot of the manga isn't identical to the novel plot, though it is similar. For example, just from volume 1 of the manga, we have the following major differences:

 When the gang is on the excursion trip at the beginning, they don't find the false minoshiro; when the queerats attack later on, Rijin isn't there to help them; and they don't get split up or taken captive during the queerat attack, and hence all five of them are present when they meet Squealer. Also, there are 3 random sex scenes featuring Maria and Saki.

Summary: if you like lesbian antics, read the manga. If not, feel free to skip it. The novel and anime are pretty much objectively superior.

Notes
1 Why them and not, say, Satoru and Shun, or Shun and Saki? Well, who do you think is reading this? (Answer: teenage boys, for the most part)
2 The Shinsekai Yori novel was a proper novel, targeted at people with excellent reading knowledge of the Japanese language. The manga, on the other hand, is replete with furigana (phonetic reading aids), which are used primarily in works targeted at people who have not yet mastered reading Japanese, i.e. school-age children.
3 See spoiler below:

 And also Reiko when she's still around.