The properties of poly(3-hydroxybutyrate-co-3-hydroxyhexanoate) P(3HB-co-3HHx) copolymers with different ratios of monomers synthesized by the wild-type strain Cupriavidus necator B-10646 on sugars, and an industrial sample from Kaneka synthesized by the recombinant strain C. necator NSDG-ΔfadB1 on soybean oil, were studied in a comparative aspect and in relation to poly(3-hydroxybutyrate) P(3HB). The copolymer samples, regardless of the synthesis conditions or the ratio of monomers, had reduced values of crystallinity degree (50–60%) and weight average molecular weight (415–520 kDa), and increased values of polydispersity (2.8–4.3) compared to P(3HB) (70–76%, 720 kDa, and 2.2). The industrial sample had differences in its thermal behavior, including a lower glass transition temperature (−2.4 °C), two peaks in its crystallization and melting regions, a lower melting point (T_melt) (112/141 °C), and a more pronounced gap between T_melt and the temperature of thermal degradation (T_degr). The process, shape, and size of the spherulites formed during the isothermal crystallization of P(3HB) and P(3HB-co-3HHx) were generally similar, but differed in the maximum growth rate of the spherulites during exothermic crystallization, which was 3.5–3.7 μm/min for P(3HB), and 0.06–1.25 for the P(3HB-co-3HHx) samples. The results from studying the thermal properties and the crystallization mechanism of P(3HB-co-3HHx) copolymers are important for improving the technologies for processing polymer products from melts.