Zero Offset relates to the zero setting tolerance during manufacture and Zero Drift relates to the expected maximum change in zero over time.
If you take a particular sensor type, each one will have a slightly different offset at zero output/reading. e.g. at 0 volts it may vary 0.02V, 0.01V, -0.05V etc…… across a batch of sensors. Some product data sheet will indicate the range of Zero Offset for a particular product, e.g. +/- 0.05V max.
All sensors are affected by environmental conditions and use over time. The output at zero reading will drift slightly over time. Some types of sensors will exhibit a greater amount of zero drift at the beginning due to settling-in period of the materials used in the construction of the sensor. Other sensors may get worse over time because the sensor performance characteristics have deteriorated due to heavier than normal use over the typical service life of the sensor.