We analyze data from the Hydrogen Epoch of Reionization Array (HERA). This is the third in a series of papers on the closure phase delay spectrum technique designed to detect the H I 21 cm emission from cosmic reionization. We present the details of the data and models employed in the power spectral analysis and discuss limitations to the process. We compare images and visibility spectra made with HERA data to parallel quantities generated from sky models based on the Galactic and Extra-Galactic All-Sky MWA (GLEAM) survey, incorporating the HERA telescope model. We find reasonable agreement between images made from HERA data with those generated from the models, down to the confusion level. For the visibility spectra, there is broad agreement between model and data across the full band of ∼80 MHz. However, models with only GLEAM sources do not reproduce a roughly sinusoidal spectral structure at the tens of percent level seen in the observed visibility spectra on scales of ∼10 MHz on 29 m baselines. We find that this structure is likely due to diffuse Galactic emission, predominantly the Galactic plane, filling the far sidelobes of the antenna primary beam. We show that our current knowledge of the frequency dependence of the diffuse sky radio emission, and the primary beam at large zenith angles, is inadequate to provide an accurate reproduction of the diffuse structure in the models. We discuss some implications arising due to this missing structure in the models, in terms of calibration, and in the search for the H I 21 cm signal, as well as possible mitigation techniques.