Knowledge of genomic features specific to the human lineage may provide insights into brain-related diseases. We leverage high-depth whole genome sequencing data to generate a combined annotation identifying regions simultaneously depleted for genetic variation (constrained regions) and poorly conserved across primates. We propose that these constrained, non-conserved regions (CNCRs) have been subject to human-specific purifying selection and are enriched for brain-specific elements. We find that CNCRs are depleted from protein-coding genes but enriched within lncRNAs. We demonstrate that per-SNP heritability of a range of brain-relevant phenotypes are enriched within CNCRs. We find that genes implicated in neurological diseases have high CNCR density, including APOE, highlighting an unannotated intron-3 retention event. Using human brain RNA-sequencing data, we show the intron-3-retaining transcript to be more abundant in Alzheimer's disease with more severe tau and amyloid pathological burden. Thus, we demonstrate potential association of human-lineage-specific sequences in brain development and neurological disease.