Assessments tend to be used in private and become public only under the assessor's control. Grades, while not truly public, are part of the administrative record available throughout an educational institution. Assessments are almost always collected in anonymous fashion and the results are released in the aggregate. Grades are identified with specific students. To use a metaphor from the calculus, assessment more resembles a partial derivative whereas grades are more recognizable as an integrative process.
Assessment tends to look at specific parts of the learning environment.
Grades are holistic in the sense that they record academic achievement for a whole project. Final grades, of course, can reduce academic achievement for an entire semester to a single mark. The subtext of a course involves the transferrable baccalaureate skills, such as critical thinking, creative thinking, writing, and analysis. Assessment findings tend to be suggestive and have pedagogical significance. That is, assessment findings shift pedagogy for reasons that need not be justified statistically, but can be justified when even one student learns better.