The following comparison is somewhat simplistic, but I hope it illuminates to some degree the different assumptions of the two approaches to assessment.
Students typically select an answer or recall information to complete the assessment.
These tests may be standardized or teacher-created. They may be administered locally or state-wide, or internationally. Behind traditional and authentic assessments is a belief that the primary mission of schools is to help develop productive citizens. That is the essence of most mission statements I have read. From this common beginning, the two perspectives on assessment diverge.