Biological phase separation forming membraneless organelles in cytoplasm and nucleus has attracted considerable attention. Liquid-like condensates are often created as spherical droplets. However, various condensates with network-like morphologies, including protein granules, localisation bodies, and centrosome assemblies, have recently been discovered in cells. Therefore, what controls the morphology of biological phase separation is a critical issue but remains elusive. Here, based on the knowledge of viscoelastic phase separation in soft matter physics, we propose that the difference in the molecular dynamics between the two phases controls the condensate morphology. Small and large mobility differences between the two phases should lead to droplet-like and network-like morphologies of the minority phase, respectively. We show that asymmetric partitioning of high-molecular-weight unstructured polymers (e.g., messenger RNA) between the two phases increases the dynamic asymmetry between the phases to form a network-like pattern of the slower phase, which may further be stabilised through inter-polymer binding.