The differences are discussed here
Export or convert raster datasets—ArcGIS Pro | Documentation
and copy raster, here
Copy Raster (Data Management)—ArcGIS Pro | Documentation
with other options covered in the Environments settings, such as
Cell Alignment, Cell Size, Compression, Current Workspace, Extent, Geographic Transformations, NoData, Output CONFIG Keyword, Output Coordinate System, Parallel Processing Factor, Pyramid, Raster Statistics, Resampling Method, Scratch Workspace, Snap Raster, Tile Size
have you tried copy raster with settings for the particular environments using arcpy (eg those that are associated with cell size, renderer etc)
... sort of retired...