Synaptic connectivity between neocortical neurons is highly structured. The network structure of synaptic connectivity includes first-order properties that can be described by pairwise statistics, such as strengths of connections between different neuron types and distance-dependent connectivity, and higher order properties, such as an abundance of cliques of all-to-all connected neurons. The relative impact of first- and higher order structure on emergent cortical network activity is unknown. Here, we compare network structure and emergent activity in two neocortical microcircuit models with different synaptic connectivity. Both models have a similar first-order structure, but only one model includes higher order structure arising from morphological diversity within neuronal types. We find that such morphological diversity leads to more heterogeneous degree distributions, increases the number of cliques, and contributes to a small-world topology. The increase in higher order network structure is accompanied by more nuanced changes in neuronal firing patterns, such as an increased dependence of pairwise correlations on the positions of neurons in cliques. Our study shows that circuit models with very similar first-order structure of synaptic connectivity can have a drastically different higher order network structure, and suggests that the higher order structure imposed by morphological diversity within neuronal types has an impact on emergent cortical activity. Author Summary We seek to understand how certain characteristics in the structure of neuron-to-neuron connectivity shape the activity of local neural circuits. The local connectivity of cortical networks features a nonrandom higher order structure characterized by the presence of tightly connected clusters of neurons. We use a biologically detailed model of neocortical microcircuitry that recreates these features and simplify its complexity algorithmically while preserving the larger scale connectivity trends. We then simulate spontaneous and evoked activity in the two models with simplified and complex connectivity and compare the resulting spiking statistics. The results allow us to characterize the role of the higher level structure of connectivity in interaction with other biological features shaping neuronal activity such as synaptic adaptation and noise.