What does Mormonism have to offer in regards to salvation that Jesus
Christ had not already accomplished?
In some sense, nothing. The atonement of Jesus Christ is what saves us. Faith in Jesus Christ is what saves us. The scriptures, however many they may be, can only lead us to Christ but never save. An illiterate person can be saved. But in this sense you have to question what any denomination has to offer, or what the visible church had to offer that made it a good idea to establish one in the New Testament. Or even what 98% of the Bible "has to offer".
However, Jesus has also said that we need to be baptized (John 3:5). In his time, he gave his apostles authority to baptize in his name. Without this authority, a baptism is just going through the motions. We believe this authority was lost early in the church history and restored by angels who conferred this authority to Joseph Smith. So in that sense, the true church should have authority to perform ordinances that Christ described as necessary for salvation.
The Church of Jesus Christ of Latter-Day Saints claims to be a restoration of the original church Christ founded. Literally the church of Jesus Christ. Whatever it was that having a church in the first place offered, is what "Mormonism offers".
So why do we need "Another Testament of Jesus Christ" when the first
testament, i.e the Bible, (specifically the New Testament) has proven
sufficient to save one's soul?
Moses was given a stick that turned into a snake to prove he was sent by God. While the Book of Mormon does restore or clarify parts of the doctrine, again, it's not the doctrine that saves, any amount of it. Think of the Book of Mormon as Joseph Smith's stick that turns into a snake.
The title changed so a significant portion of my answer doesn't really fit the title anymore. To answer the changed title, here are some of the things the Book of Mormon teaches with regards to salvation, that are disputed among the different denominations. So, in that sense, these are not "new" but really an additional witness that clarifies points and gets rid of confusion and misinterpretation. Note please the context. Joseph, in his First Vision, was told not to join any of the denominations of his day, that they were all wrong. He wasn't told "this point of their doctrine is wrong, and instead this is right". So, when he came out of that, he knew God and Jesus existed, but apart from that not really much more. The Bible is still right, but still no idea how to interpret it. He possibly knew less than what he thought he knew before. So for him when the time came to translate the Book of Mormon, the additional witness the Book of Mormon provided was very important to figure out what the right doctrine is.
Freedom of choice (as opposed to Calvinist beliefs):
2 Nephi 2: 26-27
26 And the Messiah cometh in the fulness of time, that he may redeem
the children of men from the fall. And because that they are redeemed
from the fall they have become free forever, knowing good from evil;
to act for themselves and not to be acted upon, save it be by the
punishment of the law at the great and last day, according to the
commandments which God hath given.
27 Wherefore, men are free according to the flesh; and all things
are given them which are expedient unto man. And they are free to
choose liberty and eternal life, through the great Mediator of all
men, or to choose captivity and death, according to the captivity and
power of the devil; for he seeketh that all men might be miserable
like unto himself.
Baptism is a commandment, and essential
3 Nephi 27:16 (among others)
16 And it shall come to pass, that whoso repenteth and is baptized in
my name shall be filled; and if he endureth to the end, behold, him
will I hold guiltless before my Father at that day when I shall stand
to judge the world.
... except for children (as opposed to child baptism practices)
8 Listen to the words of Christ, your Redeemer, your Lord and your
God. Behold, I came into the world not to call the righteous but
sinners to repentance; the whole need no physician, but they that are
sick; wherefore, little children are whole, for they are not capable
of committing sin; wherefore the curse of Adam is taken from them in
me, that it hath no power over them; and the law of circumcision is
done away in me.
Priesthood authority is necessary for baptism, and is conferred (as opposed to priesthood of all believers)
3 Nephi 11:21-22
21 And the Lord said unto him: I give unto you power that ye shall
baptize this people when I am again ascended into heaven.
22 And again the Lord called others, and said unto them likewise; and
he gave unto them power to baptize. And he said unto them: On this
wise shall ye baptize; and there shall be no disputations among you.
Christ saves us from our sins, not in our sins
10 And remember also the words which Amulek spake unto Zeezrom, in the
city of Ammonihah; for he said unto him that the Lord surely should
come to redeem his people, but that he should not come to redeem them
in their sins, but to redeem them from their sins.
11 And he hath power given unto him from the Father to redeem them
from their sins because of repentance; therefore he hath sent his
angels to declare the tidings of the conditions of repentance, which
bringeth unto the power of the Redeemer, unto the salvation of their
- Resurrection is unconditional (not sure who teaches otherwise, but I am sure some denomination is going to teach only the righteous are resurrected. Maybe JW)
- There is a time between death and resurrection where spirits await their resurrection (as opposed to soul sleep/ mortalism)
- the second death suffered by the wicked is being cast out from Gods presence, not a literal destruction of their immortal self
I wouldn't call any of these points new (after all, "Christ suffered for our sins so we can repent and return to God" is so simple I can literally write it in one sentence), but you have to admit these are points that are contentious among denominations and as such can't be considered to be clear beyond doubt in the Bible.