If one reads The Handbook on Formative and Summative Evaluation of Student Learning (Bloom, Hastings and Madaus, 1971) the biggest distinction (besides purpose) of formative and summative is the level of generalization. Formatives tend to be much smaller & focused, while summatives tend to be more comprehensive. A problem with trying to combine these types of evidence is that often what seperates novices from expert is the ability to work at more comprehensive levels.
For example, when studying performance of avionics technicians, Lajoe (1993) found that differences between novices and experts occurred in the over all problem solving process when cognitive load was high, such as when troubleshooting test equipment. Both groups performed well when performing discrete elements of the job, the trouble occurred when having to juggle multiple elements in the process of achieving a larger goal (p. 264).
The two graphs below are from a honors geometry teacher's grade book. The first one shows the scores for the formative type assessments (in green) and the larger more comprehensive assessments (red). The second image shows the course grade calculated using the all the grades (using the teacher's point scale) vs. just the summative assessments. The inclusion of the formatives boosts some students grades two letter grades!
Imagine if this were a group of students who haven't had a particularly good experience in math. What if they routinely missed several assignments of just performed poorly on them becasue they still needed more practice. In those cases you could see the opposite effect where the formative grade actually supressed the overall grade.