Genetic structural variation has strong potential to contribute to human evolution and disease, as large deletions, duplications, and inversions alter more base pairs of the human genome than single-nucleotide variants. However, structural variants (SVs) have remained systematically understudied because they are challenging to identify and resolve with short-read sequencing data, upon which modern genomics depends heavily. Nevertheless, targeted efforts and recent technological advances have unveiled a dynamic landscape of structural variation within the human population and between primate species, with examples tied to both genomic disease and adaptive human traits. Of particular interest is a class of SVs called segmental duplications, large blocks of sequence duplicated at low copy number that are enriched on the great ape lineage. Human-specific segmental duplications (HSDs) comprise millions of base pairs of DNA unique to our species and, intriguingly, contain genes that shape cortical development. Beyond the genes themselves, SVs can impact gene regulation by reorganizing genes and regulatory elements throughout the genome, rewiring their interactions or altering their copy number altogether. Gene regulation is itself considered a major driver of evolutionary change, and this work explores the interplay of structural variation and gene regulation to assess how SVs contribute to species-specific features in humans and other primates. We used long read sequencing, optical mapping, and single-cell template strand sequencing to discover novel SVs in chimpanzees and rhesus macaques, demonstrating that deletions and inversions in these species preserve chromatin architecture but also are associated with divergent gene regulation. In addition, we compared mRNA levels of HSD genes and their single chimpanzee orthologs and found that derived, human-specific genes are more likely to exhibit novel expression patterns compared to ancestral paralogs. To understand the molecular mechanisms underlying this divergence, we identified novel candidate cis-regulatory elements in HSDs and demonstrate that promoters and enhancers in these regions have functionally diverged in recent evolutionary time. Finally, we performed a massively parallel reporter assay to quantify the regulatory activity of thousands paralogous HSD variants, finding mostly differences of small effect, but also uncovering variants that may generate human-specific expression patterns. Taken together, these studies highlight the regulatory consequences of SVs; duplicated and rearranged regulatory elements give rise to novel expression patterns that have the potential to underlie the emergence of new traits.