aha! Now we're talking. The difference is hierarchical. Testing is a form of assessment. Assessment is the broadest term. Within assessment there are different categories: formal vs informal and summative vs formative. At least those are the big categories that I'm concerned with. As an example, think of a final exam (test) in a semester course. It would serve as a formal and summative assessment. It tells the teacher if the student has reached a predetermined level in order to advance to the next. Formative assessment, on the other hand, is concerned with learning as it is happening. For example, if I ask one of my students to reexplain a concept during class, that would be a form of summative assessment (informal too because I am not grading him or her).
Hope that clears it up a bit!