But I am wondering wouldn't the "join from a table" do the same thing?
Lookup is the way to access fields in raster attribute tables that are not value. In 9.x the syntax for this was 'raster.field' (eg 'raster.COUNT') this is now in map algebra expressed as Lookup(raster, "COUNT") If you have a active join from the raster table to another table, you can use that in the Lookup expression with the fully qualified field name in the joined table: Lookup(raster, "othertable.field")
Again in map algebra expressions must resolve to a numeric value -- you can't use map algebra to combine raster table text fields. The best you can do to combine text fields from multiple raster inputs is do a Combine, add a new text field to the Combine output raster table, add a join back to the inputs using Add Join and run Calculate Field to populate the new field from calculations using the fully-qualified related field names.