Little has a negative force, meaning almost nothing :
He has little (almost no) time to spare. I care a little for him.
A little has an affermative force and means a small quantity: I want a little sugar. I am a little tired.
The little has a double force, one negative, the other positive. It means, not much, but all that is.
'The little hope I had is gone,' means, 'I had not much hope, yet I had some least, and all of it is now gone.'
Not a little - much: He gave me not a little trouble.