They are close, but not always the same. A Net-Zero Carbon home first minimizes the use of fuels that emit carbon (or other green house gases), in favor of electric energy. Electricity can be produced with zero-carbon emissions - from clean, renewable sources like rooftop-solar. It then recommends a solar upgrade to net zero carbon emissions over a year. A Net-Zero Energy home on the other hand, optimizes upgrades that will reduce the overall energy use to net zero energy use over a year.
XeroHome currently optimizes a home for a net-zero energy goal. A future update to the tool will allow homeowners to choose a net-zero carbon goal instead.