Ok, this is a major challenge, but here's my quick investigation into the differences between Ne and Ni.
Intuition, what is intuition? One of the most useful dictionary definitions goes as follows:
3 : the power or faculty of gaining direct knowledge or cognition without evident rational thought and inference.
We can kind of stretch "cognition" to mean "formation of ideas", and "formation of ideas" to also include entertained possibilities and impressions. So the function of "intuition" could mean the immediate impressions/possibilities examined in the mind. Now, to investigate the difference between "introverted" and "extraverted" intuition, I will first need to know what the essential difference is between an introverted and extraverted function.
Functions used in an "extraverted" way have a here and now quality to them, functions used in an "introverted" way have a past, future, or timeless quality to them, and are _less_ here and now. Introverted functions are focused more towards the self, and extraverted functions focusing more outside the self into the real world.
As an introvert, one of the most obvious differences I can think of between me and someone who seems to be more extraverted than myself is their focus on the present, where as my focus is much more often on the past or future. The extraverts would argue that I place too much emphasis (or waste way too much time) on the past, and I could argue that extraverts don't exhibit the same interest of how the past affects the future, or possess the same "sentimentality" I have towards past events. With this in mind we can start to paint the differences between extraverted and introverted intuition.
"Introverted" intuition is an intuition (see original definition of intuition) that theoretically should have A. a past, future, or timeless quality to it, and B. a greater focus towards the self. That being said, it makes sense that introverted intuition is defined as having two qualities in particular, one being the entertaining of perspectives and the other being a sense for the future. A perspective is a kind of impression, or a point of view that is timeless in nature, and it can be used to examine things from an objective, "timeless" standpoint. As a primary function my hypothesis is that introverted intuitives often deal with their surrounding world by altering or fine-tuning their perspective. I know that personally, I believed very strongly that in order to succeed at something I need only acquire the correct perspective, but nowadays I realize that this in itself is only one of many perspectives.
A "sense of the future" as mentioned in many descriptions, could be the "entertaining of possibilities" that in the introverted way take on a past and future quality.
The focus towards the self is obvious, introverted intuitions deal less with the real world and tend to take on a more theoretical nature, and behaviorally, according to most type descriptions, I think it is safe to say that introverted intuitives are less likely than extraverted intuitives to be acting on their impressions in the real world.
"Extraverted" intuition is an intuition (again see original definition) that theoretically should have A. a more "here and now" quality to it, and B. a greater focus outside the self in the real world. Extraverted Intuition uses perspectives less (because perspectives have a past/future type quality to them, and are less concrete and more theoretical), and instead places a greater emphasis on the possibilities/ideas that exist in the present/real world. Also, Extraverted Intuitives seem to act on their intuitions much more often.
It's important to note that, Ni is definitely not "better" than Ne or vice versa. The extraverted intuitive might entertain an immediate real world possibility that the introverted intuitive overlooked because he or she's head was in space looking for a grand perspective on the matter (this has happened to me a lot with ENFp types). Where as, perhaps empowered by a new perspective, the introverted intuitive could open the door to a world of possibilities the extraverted intuitive would not have normally found.