This study aims to disentangle the contribution of genetic liability, educational attainment (EA), and their overlap and interaction in lifetime smoking. We conducted genome-wide association studies (GWASs) in UK Biobank (N = 394,718) to (i) capture variants for lifetime smoking, (ii) variants for EA, and (iii) variants that contribute to lifetime smoking independently from EA ('smoking-without-EA'). Based on the GWASs, three polygenic scores (PGSs) were created for individuals from the Netherlands Twin Register (NTR, N = 17,805) and the Netherlands Mental Health Survey and Incidence Study-2 (NEMESIS-2, N = 3090). We tested gene-environment (G × E) interactions between each PGS, neighborhood socioeconomic status (SES) and EA on lifetime smoking. To assess if the PGS effects were specific to smoking or had broader implications, we repeated the analyses with measures of mental health. After subtracting EA effects from the smoking GWAS, the SNP-based heritability decreased from 9.2 to 7.2%. The genetic correlation between smoking and SES characteristics was reduced, whereas overlap with smoking traits was less affected by subtracting EA. The PGSs for smoking, EA, and smoking-without-EA all predicted smoking. For mental health, only the PGS for EA was a reliable predictor. There were suggestions for G × E for some relationships, but there were no clear patterns per PGS type. This study showed that the genetic architecture of smoking has an EA component in addition to other, possibly more direct components. PGSs based on EA and smoking-without-EA had distinct predictive profiles. This study shows how disentangling different models of genetic liability and interplay can contribute to our understanding of the etiology of smoking.