Just having a brain and (most of us) two eyes gives a much more complete depiction of a given image. The twin lenses (to a camera’s single) and processing power offer binocular vision and is truly 3D.
The image you see is not on a flat surface. It is behind the flat surface, in exactly the same way that the tree is behind my window.
Ok, here’s another phenomenon I’ve noticed. When I look at myself in my bathroom mirror, I look pretty much the same as in any other mirror. I am what I am, and that’s what I see. But . . . when I look at myself in my barber’s mirror, I look positively horrible. It has nothing to do with the “infinite views” going on there, I’m just referring to what I see in the “primary” reflection. All the bad things seem to be exaggerated there. Why just in the barber’s mirror? (And no, it doesn’t matter which barber shop I’m in.)
I’ve noticed that when you view yourself in a mirror you are always looking yourself in the face and making eye contact with yourself. You get used to that and assume that’s what others see but only rarely do you look someone directly on the face with constant eye contact.
It becomes more apparent how others see you when you catch yourself in one of those three sided dressing room mirrors where you can see yourself peering in another direction. It can be very revealing and sometimes startling.
You’re right! It’s like looking at those olde timey stereo-optiscope pictures, each picture is taken from a slightly different angle, and the thing you look at them thru keeps each picture separate from each eye. When looking at a mirror, if you close one eye then the other you can sort of see that each eye’s image is slightly different due to the distance between your eyes, and with both eyes open your brain combines them and makes a stereo image (same as it does with real life!) But that doesn’t happen with a (regular) photograph (or painting or drawing or TV/movie screen or any truly 2D image)…
Not trying to be obtuse but, the image IS on a flat surface. It’s just that the flat surface is (depending on the thickness of the glass) slightly behind the surface of the glass, that the mirror is made of. 
For the OP: I have to wonder how much lighting plays a role. Bathrooms are often the best lit rooms in an entire house, designed to give even light, eliminate odd shadows, etc. Bars, on the other hand, are pretty much the opposite (intentionally dim most of the time), and a camera’s flash is often not all that flattering either. (This is why professional photographers don’t have a single flash attached to the camera - the use multiple, diffused flashes.)
This lighting can distort features because we use highlights and shadows as cues about size and perspective. Here’s just one example of that: when some girls want to look like they have cleavage, they put makeup on their breasts. The appearance of a shadow makes your brain perceive depth that isn’t really there.
I guess you could test this by taking photographs in the bathroom mirror and seeing how this compares to what you actually see.
No. The apparent image is in fact three dimensional and behind the mirror. Yes, it is reflecting off of a 2-D surface, but the transformation that happens still produces and 3-D image, and that 3-D image acts for all optical purposes like it came from behind the mirror.
I don’t know if it helps to compare it to a TV. In a TV, the image really is produced on the 2-D surface and appears 2-D. If you draw out the lines followed by the light, both the actual path and the apparent path goes from the flat screen to your eyes. But with a mirror, the image starts 3-D. If you draw the lines actually followed by the light, they hit the mirror and bounce off into your eye. Thus, the image retains all of the 3-D properties that your eyes would see in the original image. If you draw the lines of light as perceived by your eye, though, you are perceiving an image behind the mirror (at the same distance as your object).
Look at a picture of someone, and put your finger on it, covering one of their eyes (say, the left). Easy to do.
Now go look in a mirror and try to put your finger on the surface of the mirror, covering the left one of your eyes. If the image were on the mirror, this would be easy, but you can’t do it.
You can cover the left eye from the point of view on one of your eyes, but not both. If you focus on your finger, you’ll see two images of yourself, and only one will have your left eye covered. If you focus on yourself, you’ll see both your eyes, and two images of your finger.
A picture of a mirror is worth a thousand words!*
The image that you see IS reflecting off of an essentially 2D surface, but the light rays approach your eyes at the same angles AS IF they were coming from a 3D object behind the mirror.
To put it another way, the light rays you see actually are coming from a 3D object (your face, for example) and they just happen to change direction once on the way to your eyes. The image you see in a mirror is as three dimensional as if you were looking at the object directly.
*okay, so it’s a drawing and not a picture
I’ve often wondered about this myself. The best answer I could come up with is that when you view yourself in a mirror you are always seeing your self from eye-level. Photographs can be from any different angle.
A view from eye-level is generally flattering, plus, the mirror view is consistent. You see yourself from the same perspective every time and therefore become used to it whereas a photograph is a novel image every different time leading you to think things like: “I didn’t know my ear looked like THAT!”
Ask your nearsighted friend to take off her eyeglasses and look out the window into the distance. She will say she sees blurry. Now tape something like a pencil to the window glass and she can see it in focus because it is close. She can see the pencil in focus but the landscape in the background is out of focus.
Now ask her to turn around and look into a mirror placed about a foot from her eyes like a rear view mirror so she can see the same window and background in the mirror. Is the background in focus now? no, she will still see the pencil in focus and the landscape in the background out of focus.
3D. That’s what it’s all about.
Your eyes are not focusing on the reflective surface. This is also very evident if you look at a mirror which has some speck on it. Get up close and focus on the speck and the image in the background is out of focus. Now focus on the background and the speck is out of focus.
Ah-hah! I think that I’ve got it, now. Thank you all for taking the time to educate me and helping me to ‘see the light’. 
Some of the problems with photos have to do with the focal length. Different focal lengths require you to be at different distances to the subject (to take up the same space in the frame).
A visual example of this can be found here:
http://stepheneastwood.com/tutorials/lensdistortion/tilepage.htm
If you use a focal length (low) that requires you to be very close to the subject - it distorts he features and can make you look more like a chipmunk.
Photographers will sometimes refer to lenses of a certain focal length as “portrait lenses” in part due to this. I believe ~ 100mm focal length (35mm equivalent) is considered rather close to the way you look in real life.
Another anecdote. I practice life drawing by doing self portraits with a mirror. They never look quite right to Mrs. D18 until I scan and flip. Then she thinks they look like me and I think they’re a bit off!