Yea I mean intuition as Jung intended is basically having a gut feeling about things. Extraverted Intuition deals with how this manifests in the outside world. Possibilities in the outside world, basically. You can think of Ne-type as the person who is always sniffing around the corner trying to uncover something new. Many investigative journalists are Ne-types, always checking out hunches and 'instincts' about what something might possibly be or where a story might be found. A lot of sports agents would fall into this category too, being able to appraise an athlete's potential without necessarily having the real-world data to back it up (on ESPN Mel Kiper Jr, is probably an ENxP and Todd McShay is an Extraverted Sensation type, so they often come to similar conclusions but for very different reasons. Kiper is far more speculative).
As far ENFJs go, their Intuition is used in conjunction with their Feeling to typically give them a sense of the underlying implications or machinations as they relate to people. With ESFJ its much more immediate as they would often downplay their intuition, or it might even manifest more negatively, as their tertiary Ne might come out as rumors, or gossip or negative speculation (though nothing like a Si-dom). The ENFJ is a little more pointed with their Intuitions and would trust them far more. They may even have built up their sense of Intuition to the point where it rivals their dominant Feeling. This is often the case with a lot of teachers or preachers who just seem to know what a student or parishioner's needs are. They can quickly assess a situation without much outer data and come to an often accurate assessment of what the underlying issues are, where the ESFJ would tend to not do this, or be much, much more speculative (maybe getting it right sometimes, maybe not). Because the ENFJ relies on Intuitions more, their intuitive abilities will be sharper.
The other thing with ENFJ's is they can have a bit of a hedonism streak because of their Tertiary Se. You can imagine that having a sixth sense about how people work combined with a healthy gravitation toward experiencing the object, can push some of them into some pretty wild areas. Often ENFJs end up in jobs like club promoter where they thrive, both instinctively knowing how to promote, what the patrons will want and indulging in the atmosphere themselves. Where in my experience, the ESFJ is probably the drunk one trying to look cool by buying a VIP table for all his buddies, the ENFJ probably either owns the club or is throwing the party (and will know just what to do at closing to keep the good times rolling).
At their worst ENFJs can be some of the most manipulating types out there. Vito Corleone, the Godfather, is such an ENFJ, who doesn't even have to raise a finger of his own but is pulling strings many layers up from afar (that's Ni helping Fe). When he talks about making a offer they can't refuse, he already knows what the outcome will be well ahead of time, so the offer is just a pleasantry. This is typical ENFJ gone bad stuff. All the Fe social graces, but a lot of string pulling behind-the-scenes. Where the ESFJ's manipulation is usually more in your face as
@simulatedworld puts it, when talking about dom-tert looping ESFJ
Imagine someone who invites everyone over for a party, sends out a bunch of passive-aggressive texts to guilt trip anyone who didn't show up, then continually insists on leading the whole group in a variety of particular activities she s/he has planned specifically to show off how unique his/her party activity ideas are. There's no pause for internal reflection or subjective opinion on any of this; everything is designed specifically to manipulate people into giving positive feedback/reaffirming the FeNe ESFJ's overwhelming importance to the group. If anyone doesn't comply, the standard response is passive-aggressive guilt tripping designed to oust that person for being "disloyal to the group." If you don't agree with the ideas the FeNe ESFJ sees as crucial to the ethos of the group, you've already personally offended him and you're probably on his shit list now.