How I experience Fi and Ne...
Introverted Feeling: An internal knowing of what is right and wrong and what makes sense, particularly from what I call a "human standpoint", dealing with emotions & morals and how they work, and trying to harmonize the inconsistencies through evaluation. Things tend to resonate with me as truth immediately, and further reflection and analyzing sort of fits it into already existing beliefs, or susses out why it resonates as truth. Sometimes there is conflict with new info, and resolving it can distress me, as can resolving a strong emotionally-driven desire with my values. Most everything is personal in some sense....I have to relate it to myself to understand it on a level where I can decide if it is right or wrong. That relating and evaluating is not done in such a linear manner though. As mentioned, often I start with the feeling and work through the "why" afterwards. It's a bit holistic and exploratory, like starting with a lump of clay and shaping it until it's a form you can name - it's not an exact science, but an art.
There is also a welling up of feeling - not emotion, but just a feeling that says "this is me, or what I believe, or an idea I feel is worthwhile" and it can be felt in a mood, an image, a melody, a phrase, etc. When I come across a way to reflect that feeling externally, I will feel immediately, "this is how the feeling must be expressed".
Extroverted Intuition: Seeing patterns between seemingly unrelated things, applying the pattern in what seems to be unusual ways, considering many future possibilities. It jumps from A to C and then back to B and then to X and has a lot of fun along the way, discovering new ideas & formulating theories. It's a lot of "what if" and Fi tends to weed out what doesn't resonate as good to me.
I feel like Ne aids in creating the idealist aspect in myself. My feelings are made universal in the sense that the Fi ideas now go beyond me, a mere individual; it's looking at the Fi vision on a global scale and creating a metaphor for the feeling to have larger application, so it's no longer just about me. I also don't have to experience something to understand it, and it doesn't have to exist for me to imagine it - Ne extrapolates from feeling I do know and things which do exist.
Ne is a door that swings both ways - things go out through it and come in through it. It aids in shaping my Fi ideas/judgments by presenting possibilities, and it's an outlet for my Fi ideas/judgments.