Interesting question, Randy! I think the 'Clip' tool in ArcGIS did not work with raster data in the past, but it does now. There were also times when one or both of these tools clipped the data to the full extent of the mask or clip feature, rather than its exact boundaries. However I believe both tools work the same now, but Clip may run faster (See
https://gis.stackexchange.com/questions/94469/is-clip-data-management-or-extract-by-mask-spatial-analyst-more-efficient).
When working with raster data, always remember to visit the environment settings, and set the snap raster to your input raster to make sure the raster cells do not shift. Also, make sure your mask or clip feature is in the same projection as the raster.