The NSRV-CE differs from the KJV in three separate ways: the source text, the translation method, and by the addition of "gender-neutral" language.
The biggest difference is the same difference that most KJV advocates use to criticize other texts. The NRSVE-CE primarily uses the Nestle-Aland text, while the King James primarily uses the Textus Receptus text. This would be the most contended difference, and while this may not be the place to discuss the full arguments, it may suffice to understand that the KJV advocates will typically quote the superiority of their text, which is no currently used for modern English translations. They would cite the superior number of copies in agreement, and also with the concept of Divine Preservation of the Word, which they believe is applied through all times, and not just something that would happen one day. Modern advocates of the Critical, Alexandrian type texts, such as the NRSVE-CE, would argue that we have recently found some older texts, that while they are fewer in number, are more accurate. Therefore, this is one large difference in these texts.
Although I cannot cite a source regarding the translation method, from what I am seeing when I read the text, it seems that the NSRV-CE uses more of a dynamic equivalence method, rather than the formal equivalent, plenary verbal translation method of the KJV. This means that the NSRV-CE would have words or phrases made to appear more modern, and some words translated with the beliefs and understanding of the translators, rather than translating the words directly, leaving the reader to imply their intention. This also is a very common difference between translations like the KVJ compared to other modern versions.
Finally, the last difference is the inclusion of "gender-neutral" language in the NRSV-CE, which attempts to remove the gender biases of English, and possibly of the source languages. This is such as using "people" for "mankind". There are strong feelings about this change even from those who would approve of other modern translation methods as well as KJV advocates.
I did leave a couple of things out. First, I have left out the removal of the "thee/thou" pronouns, which I found mentioned in some articles talking about it. This is rather minor overall, and it seems to be tied into the translation method debate. But most noticeably, I did not mention the fact that the NRSV-CE contains the Apacrypha or Dueterocannon. That is because I feel that this is not as much of a difference as some people would imagine. The original KJV included those same books. It would be up to the reader to determine whether they accept them as cannon. Both provide them in the translations, and can be read or not read. That would be a different debate.
Personally, I am much more in favor of the KJV, and I would not rely on the NRSV-CE because of these differences. I imagine that most proponents of the NRSV-CE would reject the KJV for the same reasons. The contention from the KJV side is that if a document is not completely pure, it may contain the word of God, but it is not the word of God, which for us, is a very important distinction, as we choose to stake our souls on its accuracy. However, if one took a step back, and if they considered the Bible as they consider any other written work before the creation of the printing press, these differences are still relatively minor. If you read the full text, with all the redundancy that God built into it, it is likely that you would still come out with the same basic doctrines. Ultimately, the question of difference comes down to how accurate you need the translation to be.