Primate brains are considered to be highly similar to ours. In terms of their structure, it's more appropriate to look for the differences than the similarities.
However, you're asking for cognitive skills and these are mediated by cortical structures, and especially the higher association areas. This is exactly where the similarities are starting to break down, as the cortex, and hence higher order processing has rapidly expanded in human evolution. Notably, the surface area of the cortex has greatly expanded in humanoids through a progressive increase in the number of gyri and sulci ('wrinkles') in the cortical surface. A greater cortical surface correlates with higher cognitive abilities in humans (Fig. 1). This is what sets us apart. Cognition being
[T]he ability to interpret and connect information that is perceived in the world and to then apply this personal knowledge adaptively to future situations and problems.
While humanoids have exceptionally well-developed cognitive skills, non-humanoid primates do have been shown to exhibit quite sophisticated cognitive capabilities as well. In a review article, Hopper & Brosnan (2012) state that primates have been shown to:
- Recall serial lists of images
- Select purpose-specific tools and save them for later
- Manufacture purpose-specific tools, demonstrating an understanding of causal relations and physical properties
- Have intricate social interactions and are known to interact with and manipulate other group members
- Hopper & Brosnan. Nature Education Knowledge (2012); 5(8): 3
Fig. 1. human brain (top) and chimp brains (bottom) are very similar. Especially the cortex is larger in humans. source: Tes Teach - Brain Project