Most signals in genome-wide association studies (GWAS) of complex traits point to noncoding genetic variants with putative gene regulatory effects. However, currently identified expression quantitative trait loci (eQTLs) explain only a small fraction of GWAS signals. By analyzing GWAS hits for complex traits in the UK Biobank, and cis-eQTLs from the GTEx consortium, we show that these assays systematically discover different types of genes and variants: eQTLs cluster strongly near transcription start sites, while GWAS hits do not. Genes near GWAS hits are enriched in numerous functional annotations, are under strong selective constraint and have a complex regulatory landscape across different tissue/cell types, while genes near eQTLs are depleted of most functional annotations, show relaxed constraint, and have simpler regulatory landscapes. We describe a model to understand these observations, including how natural selection on complex traits hinders discovery of functionally-relevant eQTLs. Our results imply that GWAS and eQTL studies are systematically biased toward different types of variants, and support the use of complementary functional approaches alongside the next generation of eQTL studies.