In the polyester family, the biopolymer with the greatest industrial potential could be poly(3-hydroxybutyrate) (PHB), which can be produced nowadays biologically or chemically. The scarce commercial use of PHB derives from its poor mechanical properties, which can be improved by incorporating a flexible aliphatic polyester with good mechanical performance, such as poly(ε-caprolactone) (PCL), while retaining its biodegradability. This work studies the structural, thermal, and morphological properties of block and random copolymers of PHB and PCL. The presence of a comonomer influences the thermal parameters following nonisothermal crystallization and the kinetics of isothermal crystallization. Specifically, the copolymers exhibit lower melting and crystallization temperatures and present lower overall crystallization kinetics than neat homopolymers. The nucleation rates of the PHB components are greatly enhanced in the copolymers, reducing spherulitic sizes and promoting transparency with respect to neat PHB. However, their spherulitic growth rates are depressed so much that superstructural growth becomes the dominating factor that reduces the overall crystallization kinetics of the PHB component in the copolymers. The block and random copolymers analyzed here also display important differences in the structure, morphology, and crystallization that were examined in detail. Our results show that copolymerization can tailor the thermal properties, morphology (spherulitic size), and crystallization kinetics of PHB, potentially improving the processing, optical, and mechanical properties of PHB.