We study the distribution of the interference power in a millimeter wave (mmWave) cellular network. Such interference is random and highly dependent on the employed transmission technique, as well as the varying channel conditions and the varying association between users and base stations. Traditional networks at lower frequencies usually employ omnidirectional transmission which creates an (almost) equal amount of interference in any direction. MmWave networks, however, must employ directional beamforming transmission in order to compensate for the high path loss in mmWave frequency bands. These directional transmissions drastically change the network interference structure. We examine the interference power distributions in an mmWave network employing beamforming transmission under different user association schemes, and contrast with those under omnidirectional transmission. Numerical results using an analytical mmWave channel model and a measurement-based channel generator, NYUSIM, show that beamforming not only reduces the amount of strong interference and hence significantly enhances network throughput, but also user association can considerably alter network interference and throughput structures.