Generally speaking, there are two types of differences: differences of degree, and differences of kind.
Differences of degree occur when you have two things that belong to some relevant category, but differ in magnitude with respect to some variable shared by both things.
For example, a pin prick and a migraine headache are similar in that they are both types of pain. However, they differ with respect to the intensity of the pain.
Differences in kind occur when you have two things that don't both belong to the same relevant category. For example apples and rocks differ in kind if the relevant category is fruit.
Having set this up, how do we describe the difference between humans and other life forms? Is the difference primarily of degree or of kind?
Most everyone agrees that humans share a lot with other animals. Humans are carbon based life forms that are sustained via air, water, food. Human are born, reproduce, and die. Furthermore, humans share a lot with mammals. Humans are warm blooded, have body hair, reproduce sexually and via birth.
Some, however, go further and say that although there are lots of similarities between humans and other animals, humans still differ in kind. What does this difference amount to?
Traditionally this difference amounted to something non-physical, like a soul or a spirit. Human beings are somehow made in the image of God, whereas other animals are not. This difference is significant in that it is normative, rather than merely descriptive. According to many religious traditions, human beings are different in a way that makes them "higher" or "better" than other life forms on earth.
Nowadays there are more people who don't affiliate with a particular religious tradition. Among these, many reject the idea that there is some non-physical aspect of humanity that separates it from the rest of life on Earth.
Suppose that we accept the claim that there is no difference in kind between humans and non-human life. There's an issue that we have to deal with. We observe lots of things that humans do that other animals either don't or can't do. Here's a sample list.
Travel to outer space
Engage in nuclear warfare
Form democratic governments
What explains the fact that humans do these things and other animals don't?
One route is to say that the difference between humans and other animals is that of degree. Animals can communicate, cooperate, and build stuff. They just can't write poetry and build skyscrapers. Perhaps the difference between humans and animals is more like the difference between an adult human and an infant.
Explanation by difference of degree is kind of satisfying, but not wholly. First, the gap in intellectual ability between humans and whoever's in second place (gorillas, chimpanzees, dolphins, etc) seems incredibly vast. Why are humans so far ahead of everything else?
Second, we don't see any other species doing anything remotely close to what humans are doing. Why is it that just humans are the ones that figured out how to get to the moon? Why haven't other species developed in similar ways?