"Testing" most often measures lessons learned in a particular course, over a specific timeframe (i.e. chapter tests; midterm exams).
"Testing" usually evaluates an array of skills (define, explain, solve, analyze, design, etc.) at a time as they reflect the particular course content and the level of the course (100-level, 200-level, etc.) within the program.
"Testing" typically focuses on achievement of an individual student in a particular course.
"Assessing" generally considers a particular skill or skill set.
"Assessing" considers students' development of a particular skill/skill set over time (i.e., throughout their undergraduate studies).
"Assessing" seeks to identify areas needing improvement, with its basic notion of continuous program improvement, rather than aiming to "get an A."
"Assessing" focuses not only on learning through coursework, but in co-curricular settings, in General Education, etc. to reflect on the quality of the student's undergraduate experience.