Surface plasmon resonances in metal nanoparticles are an emerging technology platform for nano-optics applications from sensing to solar energy conversion. The electromagnetic near field associated with these resonances arises from modes determined by the shape, size, and composition of the metal nanoparticle. When coupled in the near field, multiple resonant modes can interact to give rise to interference effects offering fine control of both the spectral response and spatial distribution of fields near the particle. Here, we present an examination of experimental electron energy loss spectroscopy (EELS) of silver nanorod monomer surface plasmon modes and present an explanation of observed spatial amplitude modulation of the Fabry-Pérot resonance modes of these silver nanorods using electrodynamics simulations. For these simulations, we identify differences in spectral peak symmetry in light scattering and electron spectroscopies (EELS and cathodoluminescence) and analyze the distinct near-field responses of silver nanorods to plane-wave light and electron beam excitation in terms of a coupled oscillator model. Effects of properties of the material and the incident field are evaluated, and the spatially resolved EELS signals are shown to provide a signature for assessing Fano-like interference effects in silver nanorods. These findings outline key considerations and challenges for interpreting electron microscopy data on plasmonic nanoparticles for understanding nanoscale optics and for characterization and design of photonic devices.