Ne and Ni are supposedly the most confusing functions people try and make out. There are many false explanations I've seen before so hopefully this thread will make things clear.
Some definitions say Ni is looking through the facade and some say Ni is just knowing or having a sense about something. Personally I think it's looking through the facade. It was written that there can be no solid definition of Ni and it is more of something to be experienced. The definition I will use is the 'looking through the facade" definition. Here's an example of this Ni thought process:
"Why did that person spit to the side as he walked past me? Does he know me? Did he do it without thinking? Perhaps he spat just because he had something in his mouth and coincidentally I was there when he spat. But then again perhaps not. He could have also done it because he doesn't like the way I look, or dress, or my posture. Hmmm.. Maybe it's a sign. Maybe he's telling me something without being direct. He's wearing a similar scarf therefore he probably despised the idea of me wearing one too. Maybe he wants to see my reaction. He's not looking back so perhaps he expects me to go to him and ask him why he did that. Perhaps there is a person with a camera nearby video taping this. Maybe he is good though. Maybe he is indicating he is sick and wants me to stay away from him. Too noble. He's definitely up to something... "
It is more focused on the object.
Ne is more easily definable as the endless lines of what could happen or what could've happened. Looking at the possible situations that has or will happen in different lines of possibilities. An example of the Ne thought process would be:
"I woke up late in the morning therefore I missed the bus to work which could have had a nasty guy spitting on me which could have carried a disease which could have made me sick therefore I am happy I woke up late... but what if because I'm late my boss shouts at me and he has a disease worse than the guy at the bus and makes me even more sick and I suffer more thus making me not go to work for a longer period of time and the boss is so satisfied with my temporary replacement he fires me and I end up with no job causing me to be bankrupt and live on the streets because of the difficulty of finding a job... etc."
It is more focused on the 'could be'.
A good distinction is that Ne can be seen as a "How?" function while Ni being a "Why?" function
Ne asks "How did/could this happen?" or "How may this happen?"
Ni asks "Why did/could this happen?" or "Why may this happen?"
A detective of Ne would find a a clue in a crime scene and think to himself "How in the events did this appear here? What lead to this being here?" trying to find the story of the clue whereas a detective of Ni would look at the clue and say "Why is this clue here? What is it's purpose?" trying to find the purpose and meaning of the clue being there.
Both, however, are most likely to come to similar conclusions. Only through different thought processes.
A source in greenlightwiki has an interesting way to explain in.
"Ne looks outside the box whereas Ni looks at the box."