Vehicular platooning is a promising way to facilitate efficient movement of vehicles with a shared route. Despite its relevance, the interplay of platooning and the communication performance in the resulting vehicular network (VN) is largely unexplored. Inspired by this, we develop a comprehensive approach to statistical modeling and system-level analysis of VNs with platooned traffic. Modeling the network of roads using the by-now well-accepted Poisson line process (PLP), we place vehicles on each road according to an independent Matern cluster process (MCP) that jointly captures randomness in the locations of platoons on the roads and vehicles within each platoon. The resulting triply-stochastic point process is a PLP-driven-Cox process, which we term the PLP-MCP. We first present this new point process's distribution and derive several fundamental properties essential for the resulting VN's analysis. Assuming that the cellular base-stations (BSs) are distributed as a Poisson point process (PPP), we derive the distribution of the loads served by the typical BS and the BS associated with the typical user. In deriving the latter, we also present a new approach to deriving the length distribution of a tagged chord in a Poisson Voronoi tessellation. Using the derived results, we present the rate coverage of the typical user while considering partial loading of the BSs. We also provide a comparative analysis of VNs with and without platooning of traffic.