Are there any conflicts between how Jesus is described in the Book of Mormon vs how He is described in the Bible?

One conflict is that the teachings of the Mormons differ from some verses in the Book of Mormon. So, the question is not completely relevant. The Mormons teaching about Jesus is that both Jesus and God the Father were once men on another planet who attained godhood by being good Mormons there. This is the doctrine of eternal progression. "As man is, so God once was; as God is, so man can become."

The Book of Mormon appears to be even more explicitly Trinitarian than the Bible. However, Joseph Smith seems to have changed his doctrines after the publishing of the Book of Mormon, but perhaps embraced this earlier on. Today, the Mormons do not believe anything close to the Trinity.

So, the Book of Mormon does have some things that appear to correspond to Trinitarian theology, yet the LDS church does not hold to the Trinity.

Book of Mormon References

And now, behold, my beloved brethren, this is the away; and there is none other way nor name given under heaven whereby man can be saved in the kingdom of God. And now, behold, this is the doctrine of Christ, and the only and true doctrine of the Father, and of the Son, and of the Holy Ghost, which is one God, without end. Amen. 2 Nephi 31:21

And now Abinadi said unto them: I would that ye should understand that God himself shall come down among the children of men, and shall redeem his people.And because he dwelleth in flesh he shall be called the Son of God, and having subjected the flesh to the will of the Father, being the Father and the Son—The Father, because he was conceived by the power of God; and the Son, because of the flesh; thus becoming the Father and Son—And they are one God, yea, the very Eternal Father of heaven and of earth. And thus the flesh becoming subject to the Spirit, or the Son to the Father, being one God, suffereth temptation, and yieldeth not to the temptation, but suffereth himself to be mocked, and scourged, and cast out, and disowned by his people. Mosiah 15:1-5

Now, this restoration shall come to all, both old and young, both bond and free, both male and female, both the wicked and the righteous; and even there shall not so much as a hair of their heads be lost; but every thing shall be restored to its perfect frame, as it is now, or in the body, and shall be brought and be arraigned before the bar of Christ the Son, and God the Father, and the Holy Spirit, which is one Eternal God, to be judged according to their works, whether they be good or whether they be evil. Alma 11:44

And after this manner shall ye baptize in my name; for behold, verily I say unto you, that the Father, and the Son, and the Holy Ghost are one; and I am in the Father, and the Father in me, and the Father and I are one. . . . And thus will the Father bear record of me, and the Holy Ghost will bear record unto him of the Father and me; for the Father, and I, and the Holy Ghost are one. 3 Nephi 11:27, 36

And he hath brought to pass the redemption of the world, whereby he that is found guiltless before him at the judgment day hath it given unto him to dwell in the presence of God in his kingdom, to sing ceaseless praises with the choirs above, unto the Father, and unto the Son, and unto the Holy Ghost, which are one God, in a state of happiness which hath no end. Mormon 7:7

As well as those who should come after, who should believe in the agifts and callings of God by the Holy Ghost, which bbeareth record of the Father and of the Son; Which Father, Son, and Holy Ghost are one God, infinite and eternal, without end. Amen. Doctrines and Covenants 20:27-28

There are abundant sources on the internet about the Mormon Jesus. I'd recommend starting with this CRI page. An excerpt:

...The Bible teaches that Jesus has always existed as God (John 1:1), Mormons see Jesus as someone who worked His way up to godhood. In fact, to Mormons, Jesus is merely one in purpose with God the Father; whereas the Bible declares that the Father and the Son are also one in essence (cf. Phil. 2:6) — that they are both equally God and members of the Holy Trinity.

By simply limiting this answer to the similarities of the Book of Mormon and the Bible here is what I've put together (note this is in no way a complete compilation of the similarities but rather a starting point):

|   Doctrine                |      Book of Mormon   |      Bible (KJV)      |
-----------------------------------------------------------------------------
|                           |                       |                       |
|1. Jesus was born of Virgin|       Alma 7:9-10     |       Matthew 1:16    |
|   Mary                    |                       |                       |
|                           |                       |                       |
|2. He institued the        |     3 Nephi 18        |       Matthew 26      |
|   Sacrament for His       |                       |                       |
|   followers               |                       |                       |
|                           |                       |                       |
|3. The resurected Christ   |    3 Nephi 11:14-15   |       John 20: 26-27  |
|   had scars he showed his |                       |                       |
|   believers               |                       |                       |
|                           |                       |                       |
|4. Christ taught similar   |    3 Nephi 12         |       Matthew 5       |
|   doctrines               |                       |                       |
|                           |                       |                       |
---------------------------------------------------------------------------------

I will try to update as I find more