There is an essay at the web site Journal "First Things" that talks about defending human exceptionalism, arguing take that away and there is no such thing as human rights. After listing a series of what is considered attacks to the specialness of humans, they add to it the newest weapon; animals having souls. A story in "The New York Times" discusses the idea that cognition between humans and animals are not very different. Nancey Murphy, a philosopher at Fuller Theological Seminary, argues that all processes attributed to the mind or souls are now studied in animals. The conclusion is that there is no special creation.
The counter argument of the "First Things" writer is, "the existence or nonexistence of the soul isn’t a matter that science can measure, test, or duplicate (as a believing scientist asserts at the end of Dean’s piece)." Although this is true to a point, many of the studies that Nancey Murphy is using as a test for the soul were for centuries attributed to the the actions of the soul. This includes the ability to think; as thought was considered (and to many still is) the theological equivalant of the soul's manifestation. Pure thought (or the Word/Logos) was often considered beyond materialism, and therefore Spirit. In fact, it was God.
Humans were supposed to be made in the moral image of God, with Animals as simply food or helping to make the world beautiful. To bridge that gap between animals and humans is to put the special creation into question. Although the article opened up the possibility that the existance of the soul "whether uniquely human or present in all life" wasn't a problem, the argument mostly stands on the idea that only humans can have souls. They insist a danger in accepting animal souls.
Contrasted to this is the Mormon view that all life, especially animals, have souls of one kind or another. This is because in Mormon theology each creation is a special one, with human kind as the most special of all. From Moses 3, the spiritual creation is described:
7 And I, the Lord God, formed man from the dust of the ground, and breathed into his nostrils the breath of life; and man became a living soul, the first flesh upon the earth, the first man also; nevertheless, all things were before created; but spiritually were they created and made according to my word.
8 And I, the Lord God, planted a garden eastward in Eden, and there I put the man whom I had formed.
9 And out of the ground made I, the Lord God, to grow every tree, naturally, that is pleasant to the sight of man; and man could behold it. And it became also a living soul. For it was spiritual in the day that I created it; for it remaineth in the sphere in which I, God, created it, yea, even all things which I prepared for the use of man; and man saw that it was good for food. And I, the Lord God, planted the tree of life also in the midst of the garden, and also the tree of knowledge of good and evil.
Even though this mentions the spiritual soul of humans, plants, and animals, there is still a seperate "sphere" for each of them. Therefore, what makes humans exceptional is God's classification:
He made the tadpole and the ape, the lion and the elephant but He did not make them in His own image, nor endow them with Godlike reason and intelligence. Nevertheless, the whole animal creation will be perfected and perpetuated in the Hereafter, each class in its 'distinct order or sphere,' and will enjoy 'eternal felicity.' That fact has been made plain in this dispensation (D&C 77:3). - Church First Presidency Message, Christmas greetings, Dec. 18, 1909
For God, humans are special because we are His children. The other creations, although worthy of salvation, are not given the same exalted promises. They will give Him glory as His creations, but not to the extent that humans will by giving Him a kind of posterity. There is a gulf between humans and animals that has been divinely set. No amount of philosophy can challenge that against revelations demanding it as fact.
The idea of animals having souls is continually referenced by Joseph Smith, both in revelations and sermons. The importance of this might not be very well understood, but it shows up in many places. In one sermon Joseph Smith finds the concept of animals not having souls as a minor rejection of the glory of God by limiting His ability to save:
I suppose John saw beings there of a thousand forms, that had been saved from ten thousand times ten thousand earths like this, - strange beastes of which we have no conception: all might be seen in heaven. The grand secret was to show John what there was in heaven. John learned that God glorified Himself by saving all that His hands had made, whether beasts, fowls, fishes or rament; and He will glorify Himself with them.
Says one, "I cannot believe in the salvation of beasts." Any man who would tell you that this could not be, would tell you that the revelations are not true. John heard the words of the beasts giving glory to God, and understood them. God who made the beasts could understand every language spoken by them. The four beasts were four of the most noble animals that had filled the measure of their creation, and had been saved from other worlds, because they were perfect: they were like angels in their sphere. We are not told where they came from, and I do not know; but they were seen and heard by John praising and glorifying God.
Teachings of the Prophet Joseph Smith, p. 291
This is not to say that Mormons believe equating humans to animals is perfectly acceptable. Often times human depravity is compared to the natural lawlessness of the animal kingdom. Humans are special because they have been given a higher law to follow, and greater capacity to understand the moral implications of their thoughts and actions. To act like animals is still a sign of great sin:
Little do we realize what we have brought upon ourselves when we have allowed our children to be taught that man is only an advanced animal. We have compounded the mistake by neglecting to teach moral and spiritual values. Moral laws do not apply to animals for they have no agency. Where there is agency, where there is choice, moral laws must apply. We cannot, absolutely cannot, have it both ways.
When our youth are taught that they are but animals, they feel free, even compelled, to respond to every urge and impulse. We should not be so puzzled at what is happening to society. We have sown the wind, and now we inherit the whirlwind. The chickens, so the saying goes, are now coming home to roost.
Boyd K. Packer, “Covenants,” Ensign, Nov 1990, 84
Perhaps one of the implications to come out of the idea of animals having souls is how we are to treat them. It is well known in the revelation called "The Word of Wisdom" that meat should be eaten only in times of winter or famine. Some have speculated this was meant at the time as protection against food going bad to keep the body healthy. True as that might be, there is a companion revelation in D&C 49: 19-21 that warns against taking life and flesh wantonly. Killing of animals should not be taken lightly without spiritual condemnation.
Many prophets of the LDS Church, from Joseph Smith on, have spoken out against both cruelty to animals and over hunting. Every year the generations of warning have gone unheaded. Thousands of Latter-day Saints go into the hills and mountains to kill for sport. This isn't to say hunting is evil and should be avoided. Rather, most of the time these hunters don't take the time to respect what they kill. They don't try to protect animals from over harvest or preserve the habitat. It comes off as gluttony and violence against God's creation.
Perhaps this is the great irony of a religion that believes in both the soul of Animals and the Exalted sphere of humans. When humans don't use their higher moral capacities, all creation suffers. What seperates humans from animals is conscious choice between right and wrong. Thinking doesn't make it so, but actions do.