The Islamic tradition does not say that human beings are fully different from animals. Like animals, human beings eat, drink, move around, sleep, mate, fight, grow up, and have social communities. Allah says of animals: "There is not an animal (that lives) on the earth, nor a being that flies on its wings, but (forms part of) communities like you. Nothing have we omitted from the Book, and they (all) shall be gathered to their Lord in the end." Animals have their own ways of praising Allah just as human beings have their own ways.
However, what makes human beings special or different is not their similarities with animals but rather their differences, especially the human soul. Allah speaks of breathing some aspect of the divine spirit into the human being. As a result, the human being has free will, a strong intellect and capacity for abstraction, can develop or devolve spiritually to very high or low levels, can rise higher than the angels, and can manifest a variety of the names of Allah. It is the human being that took on the risk/responsibility for these things, as mentioned in the Qur'an, whereas animals did not. Therefore there is a difference between the human being and other animals, even though there are also some similarities.
Otherwise one could also say that human beings are similar to piles of dirt, because all the elements that make up the human being come from the earth, and yet it is this quality of life and soul that make a human being different from a pile of elements.
How much this relates to DNA is an open question, but it is my own understanding that at least some of what distinguishes the human soul is not material in origin and does not come from DNA.