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I´m wondering about the same question during my current work. The same difference occurred, when I ran both tools. Did you do a supervised classification? In my case I classified the maximum likelihood. By classifying with this tool,
you include a set of values to one class, while otherwise in zonal statistics the value of all pixels below your mask (shapefile) is estimated. That might cause the visual difference in your result. But I can´t answer the question,
whether a classification or pure zonal statistics is more reliable.
Maybe someone else knows about the purpose and background, both of these tools are applied and interpreted.
The classification statistics box on the Classification dialog (on the classified renderer) are an estimate. Not all of the values are used when the input is floating point which typically a density raster would be. It would be too slow. If you want to have more accurate stats in this dialog you can follow the steps outlined in one of our knowledge base documents.
HowTo: Compute the histogram with all the unique values, when using the classified renderer
Johannes- Zonal Stats does not estimate the value of all pixels below your mask. Maximum Likelihood wouldn't be used on a Density surface. Please describe the issue you are seeing with your data.