Distribution networks-from vasculature to urban transportation pathways-are spatially embedded networks that must route resources efficiently in the face of pressures induced by the costs of building and maintaining network infrastructure. Such requirements are thought to constrain the topological and spatial organization of these systems, but at the same time, different kinds of distribution networks may exhibit variable architectural features within those general constraints. In this study, we use methods from network science to compare and contrast two classes of biological transport networks: mycelial fungi and vasculature from the surface of rodent brains. These systems differ in terms of their growth and transport mechanisms, as well as the environments in which they typically exist. Though both types of networks have been studied independently, the goal of this study is to quantify similarities and differences in their network designs. We begin by characterizing the structural backbone of these systems with a collection of measures that assess various kinds of network organization across topological and spatial scales, ranging from measures of loop density, to those that quantify connected pathways between different network regions, and hierarchical organization. Most importantly, we next carry out a network analysis that directly considers the spatial embedding and properties especially relevant to the function of distribution systems. We find that although both the vasculature and mycelia are highly constrained planar networks, there are clear distinctions in how they balance tradeoffs in network measures of wiring length, efficiency, and robustness. While the vasculature appears well organized for low cost, but relatively high efficiency, the mycelia tend to form more expensive but in turn more robust networks. As a whole, this work demonstrates the utility of network-based methods to identify both common features and variations in the network structure of different classes of biological transport systems.