This is just the way I would explain the difference, but I am no dictionary so here we go:
Ignore is an active choice, if somebody is doing something and you do not want to acknowledge it you would ignore them.
Neglect is when you have some kind of responsibility and you choose not to do it, we talk about people neglecting children if they don't feed them, or bathe them or stuff along those lines.
Disregard is to basically choose to not let something bother you, synonym would be to not pay attention to something.
Overlook is when you see something that is wrong and you choose to not correct it, like a corrupt police officer may overlook some crime in exchange for money.