What, if anything, sets humans apart from other species? As I grew up, I was taught that the difference between humans and animals was that human beings have free will and animals do not. The basis is that animals will act accordingly to the nature of their species. I never fully understood this statement and still wonder whether it is a valid claim.

From an evolutionary  perspective, we would be regarded as a species at the top of the food chain. The main advantage in our development is our brains.

If we put aside the differences in what human beings are capable of, e.g. making clothes, machinery, etc., is there a real distinction between human beings and other animals? Do we really have free will, or an illusion of free will? Or are we behaving, as other animals, in accordance with the nature of our species?

I know this is a big question and could result in a range of debates over my definitions. So for the sake of clarity, let's keep this question limited to what defines a human brain as opposed to an animal brain and any evidence that supports any significant difference, allowing for the fact that the human brain is well-developed (as opposed to, e.g. to our canines).
Some articles with some information pertinent to your question, hope this helps:

According to "Human and animal cognition: Continuity and discontinuity" (Premack, 2007), that as several species have skills once thought to be unique to humans, that this suggests that there is a misleading (the article's word) disparity between 'brain' and 'mind'.  So the article's focus is on:


  What are the dissimilarities?


When examining 8 aspects, or competencies, including teaching, short-term memory, causal reasoning, planning, deception, transitive inference, theory of mind, and language, the authors found that the similarities to be minor, but the dissimilarities to be very large and that there is no disparity.

Each of these points are discussed in depth in the article (too big for this answer); however, the article's author concludes with a fundamental difference:


  Animal competencies are mainly adaptations restricted to a single goal. Human competencies are domain-general and serve numerous goals. For instance “planning” may be tied to episodic memory, suggesting a broad competence. 


Going further, according to "Evolution of the brain and intelligence", (Roth and Dicke, 2005), that a main difference is due to:


  factors that correlate better with intelligence are
  the number of cortical neurons and conduction velocity,
  as the basis for information-processing capacity.


and critically (bolding mine):


  The outstanding intelligence of humans
  appears to result from a combination and enhancement
  of properties found in non-human primates, such as
  theory of mind, imitation and language, rather than from
  ‘unique’ properties.