I have no idea about the completeness but I think there is more involved than data completeness. I use GeoNames as they have a Creative Commons By Attribution license. Depending on where you want to use the data it may/may not be more convenient to use one or the other.
GeoNames has better global coverare for cities and other populated places whereas google has coverage for street level geocoding in some countries, something GeoNames is missing completely.
Depending on your focus either GeoNames or Google is more complete.