We present new multifrequency radio continuum imaging of the dwarf starburst galaxy NGC 625 obtained with the Very Large Array. Data at 20, 6, and 3.6 cm reveal global continuum emission dominated by free-free emission, with only mild synchrotron components. Each of the major H II regions is detected; the individual spectral indices are thermal for the youngest regions (showing largest Hα emission) and nonthermal for the oldest. We do not detect any sources that appear to be associated with deeply embedded, dense, young clusters, although we have discovered one low-luminosity, obscured source that has no luminous optical counterpart and resides in the region of highest optical extinction. Since NGC 625 is a Wolf-Rayet galaxy with strong recent star formation, these radio properties suggest that the youngest star formation complexes have not yet evolved to the point where their thermal spectra are significantly contaminated by synchrotron emission. The nonthermal components are associated with regions of older star formation that have smaller ionized gas components. These results imply a range of ages for the H II regions and radio components that agrees with our previous resolved stellar population analysis, where an extended burst of star formation has pervaded the disk of NGC 625 over the last ~50 Myr. We compare the nature of radio continuum emission in selected nearby dwarf starburst and Wolf-Rayet galaxies, demonstrating that thermal radio continuum emission appears to be more common in these systems than in typical H II galaxies with less recent star formation and more evolved stellar clusters.