I IZ HONORED TO BE TAGGED.
Alright so E4 and Fi may look similar, and often fit together for types using Fi, but they are different.
First let's look at E4.
Heart types in enneagram have a very core problem with feeling "worthless" in some way, and each of them handle this sort of shame differently. 3s tend to try to be more successful admirable people to combat it. 2s make themselves helpful or even outright "seductive" to gain worth from love.
4s find worth in their individuality.
Some 4s will try very hard to be different, not molding into the crowd. Underneath, 4s are actually envious of those who get more than they do or fit in with the crowd, or feel "defective" because they are different. Rather than confront this, they deal with it by (depending upons wings and instincts) will become self-loathing, withdraw, overly-competitive, aggressive, nitpicky of imperfections.
4s can find beauty within suffering, and some 4s in a sense have a masochistic relationship with suffering and enduring. They feel taking on their sufferings and embracing them is what makes them "special" and "worth something."
Now lets look at Feeling functions.
It must be generally divided again how easily Fe and Fi are mistaken based on how someone chooses to express their emotions. Fi is not more or less emotional than Fe. Fi can talk about their feelings, like an Fe. Fe can hide their emotions, like an Fi can. Emotions is NOT THE CORE OF FEELING FUNCTIONS.
Feeling functions are based upon your ethics, how you got them, how you adjust/keep them, and how you handle the world around you as to your ethics.
Fe and Fi both have personal ethic systems. Stereotypical views of Fe makes them sound as if they are groupthinkers, which I suppose an unhealthy or immature Fe could subscribe to this, but we should not define types by unhealthy or immature types.
Introverted functions vs. Extraverted functions:
I feel this is also important to state. You could be an outgoing Fi as much so as an Fe. Your social style does not necessarily play into how you use functions.
An introverted function is one where you need to spend more time processing things. It is also an internalized function. It does not need an external consensus and may not even seek an external consensus. It tends to go by its own terms and discoveries.
An extraverted function is one where you process things through the whole. It is an externalized function, and ones you often show to the world. Often these functions may desire validation of their ideas from those close to them, or from studies, or something as that. It has its own terms and discoveries, but may also adjust them according to consensuses they seek.
So, this was important to discuss because now we're getting into what makes Fi, Fi. And what makes it different from E4.
Fi is internalized ethics, in that sense. Fi values keeping to their values. Fi develops its consensus through its own process. They may need more time to examine an issue before having an ethical opinion on it, because they must decide if it fits their internal code of ethics. They are more likely than Fe to associate with many different others, because they're not as focused on the "external peace" but on "inner peace." Fi can be as empathetic as an Fe. Fi can have as strongly spoken values as Fe. The difference is how the ethics are developed.
So they sound a little similar, in the form that both E4 and Fi are focusing on an "inner peace" they're desiring. However, Fi does not necessarily makes you envious of those who appear in a superior position, or make you feel defective. Fi does not mean you're a little special thing. Fi just means you gather your ethical system through an internalized system, rather than an external one.
E4 is about dealing with a feeling of being worthless or defective. E4 does this by desiring to be different or special. An E4 may have an Fe or Fi value system, but push for something different in the name of being "different" so they can feel "special." On stackings, some 4s may even appear more like 8s, or 1s.
Hope this helps some. I feel like I said a lot and not sure I got the whole point in... XD