I'm having difficulty understanding what Fi is, and every time Fi vs. Fe is discussed in the NF forum, the distinctions get messy because they take every attempt at definition as personal attack (and in many cases, they are personal attacks).
So, For ENTP/INTP/ENTJ/INTJ, what do Fi and Fe actually do? How would I more easily recognize these functions in NTs in every day life?
Would Fi mean you care that Woody Allen married his daughter? Or would Fe care about that? Or would they care for different reasons? (Fe because it's considered destructive to society? Fi because ...? I don't know what Fi is so I don't know how to finish that, and I'm not certain I got Fe right, but I tried to imagine why I might care about that, and that was what I came up with. I don't care, by the way.)
So anyway ... a little help?
Whether an NT uses Fe or Fi depends upon whether that NT uses Te or Ti.
Fe always pairs with Ti (for NTPs) and Te always pairs with Fi (for NTJs.)
As for how they actually make decisions, the basic difference between introverted and extroverted functions is that introverted functions are self-validated and operate entirely according to one's personal perspective with no input from the outside world. The user of an introverted function simply
knows through direct experience whatever it is that that function tells him is important.
Extroverted functions, on the other hand, change and adapt according to present external conditions. In the case of Fi vs. Fe, this amounts to the following:
Fi defines ethics according to the self. What is right and wrong is a function of private and personal values, and does not depend at all upon any validation from other people or any external factors at all. Introverted functions operate "in a vacuum"; that is, an Fi user is concerned with what is inherently ethical as defined by his own internal value set, and this is very very unlikely to change.
"What do I know in my heart is right regardless of what anyone else says?" Fi is more likely to declare its enemies evil or amoral.
Fe defines ethics according to the environment. What is right and wrong depends upon what ethical/social standards are observed and practiced by the community group. Fe adapts its conception of ethics to the emotional texture of its environment; this is why strong Fe users are so attuned to the emotional needs of others--the validation comes from without, not from within. To Fe, there is no such thing as "ethics in a vacuum" because we cannot make an external judgment until some external goal or context has been defined.
"What does my family unit or other cooperative social group hold as the most important ethical values?" Fe is more likely to declare its enemies rude or inappropriate to the current social context.
As for Woody Allen marrying his daughter, both Fi and Fe could have serious problems with this in the right context, or they might both be ok with it, depending on personal factors.
If Fi had a problem with this, it would be due to a rigid internal personal belief that marrying your daughter is morally wrong, period. Not everyone's Fi holds the same moral values, though--the key is where the source of moral decision comes from.
If Fe had a problem with this, it would be due to the fact that this marriage violates community social and ethical standards in the group by which the Fe user defines his place in relation to others. "Our society doesn't tolerate this" would be an Fe-oriented objection; however, an Fe user might also be perfectly fine with this if he were operating within a social group that doesn't consider this to be a problem.
An Fe user moving from one culture to another would have an easier time adapting his etiquette standards to fit the new surroundings--Fi would almost never do this, as it doesn't value any external validation for its moral belief system.
"What would Fe and Fi each think about this particular moral issue?" is not really the right question. Either one might find any given issue moral or immoral; there are no moral beliefs that are 100% universal among all Fi (or all Fe) users.