Background: I am a mathematician. The golden ratio certainly exists mathematically, it does appear on occasion in nature (though not as often as people think) and when it does occur then there are proper scientifically falsifiable theories as to why it occurs (the spirals on a pinecone are one example, I believe, though the spirals on a nautilus are not). However, it is equally well known that if you take enough different measurements then you will find any number or ratio that you want to find. Simply finding the golden ratio somewhere is not anything exciting. Explaining why it should be there is the important thing.

As to its presence in art, I have nothing to say. I'm a mathematician, after all.

But that's by-the-by and has already been adequately covered in all of the other answers. What I haven't seen yet is an image with the direct comparisons. So here is one. In the upper picture, the green lines are a third of the way in from the top left corner, the "golden" lines are the appropriate version of the golden ratio in from the top left corner. As many have said, there ain't a lot in it.

The bottom picture refers to something I'm a little surprised hasn't been mentioned. That is that the sensor size, and the size of the "standard" photograph, aren't in the correct ratio for the golden ratio to be correctly applied! For a photograph to be taken seriously as "having the golden ratio" property, then I'm afraid that it would need to be sized in the ratio of 1:1.6180... The standard photograph should thus be 4 inches by a shade less than 6.5 inches.

Speaking purely speculatively, and ignoring the "nature versus nurture" aspect of this, I'd say that the fact that standard photographs are 6x4 says that we tend to prefer things that are a bit rectangular, but not too much, and so I would go along with a theory that placing the object of interest appropriate to the picture size makes for a more pleasing picture. But the fact that we're happy with 6x4 pictures and there isn't a huge clamour for 6.4721x4 pictures shows that we're actually not that fussy and can't actually tell the difference between 0.666666 and 0.6180

Here's the pictures: