What, if anything, sets humans apart from other species?
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.