Which is better: writing a traditional cover letter or an "actual" letter?

I will be soon applying to the department of computer science of numerous top universities for PhD studentship soon.

There is a single question in my mind; wheter to write a cover letter or an actual letter.

Some of the professors argue that a cover letter should be as formal as it can get while others think that writing a candid letter is better.

Honestly, I have been a student who is far from being accepted if considered my average grades (2.85/4.00 in BSc 3.50/4.00 in MSc). Similar to every unsuccesful student, I have my own reasons.

Should I write a traditional cover letter that formally states the reasons that I apply, the achievements, and the formal purpose? Or should I write what I feel, my reasons for being unsuccesful, not publishing a paper (except two papers to two bogus conferences), during MSc and my feelings?

Obviously, it is dependent on the commitee in charge. However, I am sure that there are many professors out there that experienced similar things.

P.S. I know that I have to lower my standards. Assume that I cannot.