To address the question of whether the so-called ultracompact dwarf galaxies (UCDs) are the remnant nuclei of destroyed early-type dwarf galaxies (dEs), we analyze the stellar population parameters of the nuclei of 34 Virgo dEs, as well as 10 Virgo UCDs, including one that we discovered and which we report on here. Based on absorption-line strength (Lick index) measurements, we find that nuclei of Virgo dEs have younger stellar population ages than UCDs, with averages of 5 Gyr and >10 Gyr, respectively. In addition to this, the metallicity also differs: dE nuclei are on average more metal-rich than UCDs. On the other hand, comparing the stellar population parameters at the same local galaxy density, with UCDs being located in the high-density cluster regions, we do not find any difference in the stellar populations of dE nuclei and UCDs. In those regions, the dE nuclei are as old and as metal-poor as UCDs. This evidence suggests that the Virgo UCDs may have formed through the stripping of dE nuclei.