This is not scientific but I will tell you how I interpret the question. Weakness is when a muscle is not strong. The muscle can simply not pick up a lot of weight.
Fatigue is when the muscle does not have a lot of endurance. It may be able to lift a massive amount, but it gets tired easily.
Both are important. Think about pushups. I have seen huge body builders that could not do 20 pushups because there muscles fatigued so easily. On the flip side, I have seen cross country runners who could use their legs for miles but not squat 200 pounds. You have to train for both.