For PROCESSING errors, the Return PARAMETERS for BAPI should be used.This parameter returns exception messages or success messages to the calling program.
BDC:
BDC goes through all the screens as a normal user would do and hence it is slower.
Background and Foreground processing options are available for BDC.
BDCS would be preferred for large volumes of data upload since background processing OPTION is available.
Errors can be processed in SM35 for session method and in the batch input program for Call Transaction method.