Difference between Not half and Not the half

not half —(coll.)

1. not nearly; not at all:

• But why don’t you come to my room? I have some Scotch—not half bad.

2. (emphatic) only a fraction or a small part of smth.:

• So, I have to read that and write about it. And that’s not half of my homework.

not the half — = not half 2:

• He accused them of being responsible for the error, and that’s not the half of the story.