Both versions use distinct sets of manuscripts for the Old Testament and New Testament.
One might say that the NIV uses an "older" Old Testament manuscript on occasion by deferring to the Septuagint or Dead Sea Scrolls (as explained below), but I am not sure this is significant. There are rumors that the King James translators may also have done likewise, even if forbidden by the translation rules the King had put in place (see, e.g., Adam Nicholson, The Making of the King James Bible).
In the case of the New Testament, whereas the King James editors seemed to have consulted a single 1598 manuscript compilation, the NIV, which relies on the so-called "Critical Text", probably included some "older" manuscripts in its translation. But it is very difficult to judge whether the underlying text is older or newer from the age of the manuscript. A newer manuscript may actually hold a copy of a variant that had somehow been lost, for example.
With regard to archaic English language, some prefer the King James Version because it preserves a difference between singular and plural forms of the 2nd person - something now lost to the English language, but is present in the underlying Greek. This distinction has been lost in modern English (except, perhaps, in Texas, where "y'all" is used), but it was present in Jacobian English (i.e. "thou" for you singular, "ye" for you plural. There is a discussion of this here.
The Introduction provided in the 2011 New International Version states:
For the Old Testament the standard Hebrew text, the Masoretic Text as
published in the latest edition of Biblia Hebraica, has been used
throughout. The Masoretic Text tradition contains marginal notations
that offer variant readings. These have sometimes been followed
instead of the text itself. Because such instances involve variants
within the Masoretic tradition, they have not been indicated in the
textual notes. In a few cases, words in the basic consonantal text
[the original Hebrew used no vowels] have been divided differently
than in the Masoretic Text. Such cases are usually indicated in the
textual footnotes. The Dead Sea Scrolls contain biblical texts that
represent an earlier stage of the transmission of the Hebrew text.
They have been consulted, as have been the Samaritan Pentateuch and
the ancient scribal traditions concerning deliberate textual changes.
The translators also consulted the more important early versions—the
Greek Septuagint, Aquila, Symmachus and Theodotion, the Latin Vulgate,
the Syriac Peshitta, the Aramaic Targums, and for the Psalms, the
Juxta Hebraica of Jerome. Readings from these versions, the Dead Sea
Scrolls and the scribal traditions were occasionally followed where
the Masoretic Text seemed doubtful and where accepted principles of
textual criticism showed that one or more of these textual witnesses
appeared to provide the correct reading. In rare cases, the committee
has emended the Hebrew text where it appears to have become corrupted
at an even earlier stage of its transmission. These departures from
the Masoretic Text are also indicated in the textual footnotes.
Sometimes the vowel indicators (which are later additions to the basic
consonantal text) found in the Masoretic Text did not, in the judgment
of the committee, represent the correct vowels for the original text.
Accordingly, some words have been read with a different set of vowels.
These instances are usually not indicated in the footnotes.
The NIV Old Testament omits the Deuterocanonical books that were part of the early Church Old Testament Canon (usually referred to by Protestants as "Apocrypha"). These books were also included in the 1611 King James Version, as well as in other versions based on the Latin Vulgate (i.e. Douay-Rheims) and Septuagint (e.g. the 1851 English translation by Sir L.C.L. Brenton).
The original 1611 King James Version and subsequent updates published by Oxford and Cambridge in the ensuing centuries included the Deuterocanonical books, which were written in Aramaic and Greek. I have never seen anything identifying which particular manuscripts the translators consulted for these. As far as I know, only Cambridge continues to publish an edition of the King James Version with the Deuterocanonical books included.
The underlying Hebrew text is supposed to be a version of the Masoretic Text compiled by the Tunisian Jew of Spanish origin and later Christian convert Jacob ben Hayyim ben Isaac Ibn Abonijah, published by Daniel Bomberg in Venice sometime around 1525 (Introduction to the Rabbinic Bible, tr. Christian Ginsburg, p. 2-7).
Again, according to the Introduction in the 2011 NIV:
The Greek text used in translating the New Testament is an eclectic
one, based on the latest editions of the Nestle-Aland/United Bible
Societies’ Greek New Testament. The committee has made its choices
among the variant readings in accordance with widely accepted
principles of New Testament textual criticism. Footnotes call
attention to places where uncertainty remains.
The Greek New Testament the authors refer to is a compilation of hundreds of different Greek manuscripts. The editors essentially judged all of the variants available for each verse and made a decision as to which particular variant reading to select. Bruce Metzger has published separately a Textual Commentary that explains each decision made. The resulting text is sometimes referred to as the "Critical Text".
Dr. Maurice Robinson claims in the introduction to a modern edition of the 1550 Stephen's Textus Receptus that there are actually several extant Greek texts published around that time frame that are similarly named. He writes:
The Stephens 1550 edition of the so-called “Textus Receptus” (Received
Text) reflects a general agreement with other early printed Greek
texts also (erroneously) called by that name. These include editions
such as that of Erasmus 1516, Beza 1598, and (the only one actually
termed “Textus Receptus”) Elzevir 1633. Berry correctly notes that “In
the main they are one and the same; and [any] of them may be referred
to as the Textus Receptus” (Berry, p.ii).
All these early printed Greek New Testaments closely parallel the text
of the English-language Authorized (or King James) Version of 1611,
since that version was based closely upon Beza 1598, which differed
little from its “Textus Receptus” predecessors. These early Greek “TR”
editions generally reflect (but not completely) the “Byzantine
Textform,” otherwise called the “Majority” or “Traditional” text,
which predominated throughout the period of manual copying of Greek
New Testament manuscripts.
Dr. Robinson also explains the key differences between the Critical Text (e.g. NIV) and the Textus Receptus (KJV):
The user should note that the Stephens 1550 TR edition does not agree
with modern critical editions such as that published by the United
Bible Societies or the various Nestle editions. These editions follow
a predominantly “Alexandrian” Greek text, as opposed to the Byzantine
Textform which generally underlies all TR editions. Note, however,
that 85%+ of the text of ALL Greek New Testament editions is
He also points out that the New King James Version (NKJV), published by Thomas Nelson, footnotes verses where the CT and TR variants diverge.
I would also point out that the Greek text that is more or less "official" for the Greek Orthodox Church is the 1904 Text of the Ecumenical Patriarchate of Constantinople. It generally agrees more closely with the TR and Majority Text than with the CT. Laurent Cleenewerck's Eastern Orthodox Bible: New Testament is a translation of this text into modern English. In each case where the Patriarchal Text diverges from either the TR, Majority Text, or CT, he shows the divergence in a footnote.
[Note: The above is essentially the answer I provided to the same question on the Christianity.SE forum. This question is very old here and has had no one answer it, so I am volunteering the above on this site as well.]