Weaknesses are something that creep out from all the dark places of your psyche when editing. The wish to be perfect is most severe in these moments. Overall, I don't mind my weaknesses because I understand them, they don't hide as faceless bogeymen around the corner waiting to undercut my newest story. By knowing your weaknesses, you can empower your work because it gives you something to improve upon and it won't surprise you when critiquers mention them.
My biggest weakness when writing is my description. I've done a lot to improve my description, and in the world of fantasy, description should be a writer's strong point, but my ability to show a scene will always be lacking. I accept that, and do my best to do what I can to portray the events and characters in my story as vividly as I am able.
My biggest strength (and this is something you need to discover, so that when you are thinking about your weaknesses you can at least smile about what you do well), I think, is my dialogue. I feel like I have knack for funny conversations, and for meaningful/thought provoking statements.
Ultimately, by discovering your weaknesses you can find a place to improve your work and by locating your strengths you can find the elements of your writing which keep you going even when your writing is lacking; those elements that remind you that you have something to offer, that there is something you do well and there is a reason to carry on writing, no matter how poorly something looks or how brutally something is critiqued.