I think photography is one of the most intensive art forms - where the viewer can see so much of who the photographer is just by looking at their work. It is essentially a physical depiction of how the photographer see's the world - an insight to how their mind, experiences, abilities all meld together. It is intensely personal.
While I do think that some people have just a natural interesting viewpoint in the world, I also think it is definitely something that can be taught and learned if the photographer wants to put the time and effort into retraining their brain to look at different things in this world.
There are a lot of differences between a portrait (be it person, place or thing) and a snapshot. A lot has to do with technical ability, the ability of the photographer to show what they find interesting about that particular subject, the personality of the photographer to bring out the personality of the subject, etc. It has been my experience, while photographers are intense people, most are extremely charismatic and have an 'it' factor about them.
Anyway... there is a definite difference between a snapshot and a portrait. I think a portrait is well thought out on many different levels. Thinking something through doesn't necessarily take hours (although some do take hours and days of planning and patience), it can be a snap-decision that something is portrait worthy. But none the less, the photographer has seen that moment and had the technical skill to pull that moment into a stunning portrait. These little moments happen a 1000 times a day - the photographer has just tuned themselves into finding those moments.
Snapshots are taken to document people, place, thing and don't necessarily have the time and thought process put into them. A lot of times, snapshots are forced both in terms of subject and/or technical aspects.
I think the more time you cultivate taking a portrait, the more your imagery results in that endeavor. When I take a picture, 8 times out of 10 I feel confident that I have a stunning portrait to show someone. But that is because I've put years of practice and thought into that portrait, it's something I practice every single day, and an eye that I've cultivated. While I think I might have some kind of natural eye (not much, but a little), I do continually ask myself - WHY am I taking this picture? WHAT am I trying to say? HOW do I intend on displaying this to the world? WHAT can I do to make this more interesting or different? If I can't answer those questions, I don't necessarily take the shot.
I had an interesting experience on Memorial Day... I had been shooting a lot that particular weekend and I just wasn't in the mood to take my gear along with us on a family outing. So instead, I pulled out my sorely neglected D80 and stuck it on Auto. I don't think I've used Auto in about 5 years... Anyway, I stuck it in Auto with the express decision that I was just going to take snapshots that day. I wasn't going to think about ANYTHING in terms of exposure, composition, or any artistic form what so ever. I was just going to shoot just to shoot. It was fun and liberating. I came home with a bunch of snapshots of the day - but very liberating to not think for once. While it was liberating... I actually had to think about not thinking... and ironically, I came home with an amazing portrait of my father in law playing with my daughter. Because I just couldn't turn that part of my training off!