Zika virus (ZIKV), which is mainly transmitted by Aedes albopictus in temperate zones, can causes serious neurological disorders. However, the molecular mechanisms that influence the vector competence of Ae. albopictus for ZIKV are poorly understood. In this study, the vector competence of Ae. albopictus mosquitoes from Jinghong (JH) and Guangzhou (GZ) Cities of China were evaluated, and transcripts in the midgut and salivary gland tissues were sequenced on 10 days post-infection. The results showed that both Ae. albopictus JH and GZ strains were susceptible to ZIKV, but the GZ strain was more competent. The categories and functions of differentially expressed genes (DEGs) in response to ZIKV infection were quite different between tissues and strains. Through a bioinformatics analysis, a total of 59 DEGs that may affect vector competence were screened—among which, cytochrome P450 304a1 (CYP304a1) was the only gene significantly downregulated in both tissues of two strains. However, CYP304a1 did not influence ZIKV infection and replication in Ae. albopictus under the conditions set in this study. Our results demonstrated that the different vector competence of Ae. albopictus for ZIKV may be determined by the transcripts in the midgut and salivary gland, which will contribute to understanding ZIKV–mosquito interactions and develop arbovirus disease prevention strategies.