Why Are Animals Different from Humans, According to the Bible?

I think the best answer is to tell the story of God's desire to create people that he can be in relationship with. He made the world with a desire to have a special relationship with people that He would make in His own
image. The idea isn't that He doesn't care about animals or anything else that He has made. Rather, the idea is that in His infinite creativity, God wanted to share as much of Himself as He could with us. 

So human beings were the pinnacle of His creation, the highest form of life that He created. This is what it means that we were made in His image. He made us to have the capacity to know Him and to relate to Him. One picture that the Bible gives of the relationship between God and human beings is that of a father to his
children. God wanted to create a world where people could relate to Him as family, and He created the rest of the world to be the environment where that relationship could exist and flourish. So the world God created teaches us all about Him and what He's like. The world also shows us how we can live lives that would imitate Him. I believe this is why He created so many forms of life that aren't in His image. These other forms of life are gifts from God for us to enjoy and take care of and to use. These become an opportunity for us to reflect God's care and stewardship--celebrating what is good and using what is useful. All that we do in relationship to the rest of creation is to bring about abundant, teeming, fruitful, and flourishing life. God gave us a world that was very good, and our stewardship of the world was designed to make it even better.

So in relationship to other forms of life that we are stronger and smarter than, we get a very sincere opportunity to taste a little bit of what it's like to be God. All that we do should reflect God and how He treats us (with appropriate differences and qualifications).

Very practically, animals are gifts from God to us so that we could learn more about Him and be reminded of what He is like. They are gifts given to us to help us flourish. They are opportunities for us to learn how to be
good stewards and increasers of multiplying life.