Alright. . . it's go time! . . . just kidding.
First, a little background:
I was originally tested around 21 by a friend of mine who was in a psych PhD. program and wanted to use me as a testing guinea pig for her Jungian theory class. And after a very long test and a series of oral questions . . . she declared that I was an INTP, told me what that meant, and I filed it away for a few years - although it struck me as an excellent explanation for why I am the way I am. Additionally, I recall feeling relieved that there were other poor souls that were suffering the same fate as I.
A couple of years past, then ran into typology again via a book. I took the test at the end of the book. . . and scored an INTP with a weak P. Next stop, I was working on a research paper where I was attempting to make a connection between personality types and political positions (along with a few other things), found the site, took several tests. . . started scoring either INTP with a weak P, or INTJ with a weak J. And that trend has continued up to the present.
Last bit of information: When I take the Cognitive Processes test, my P/J has fallen 51/49 to the J-side the last few times. And - this probably explains my interest in playing sports throughout my life - I'm a very strong N, I, & T, but I have a highly developed Si and Se as well.
With all of that, I bid you luck. Because I believe the answer is complicated, although the question seems rather simple: What is the difference between an INTP with a weak P and an INTJ with a weak J?
Good luck. And thanks for your help.