There are things that regular photography can't capture.  I don't think that regular camera lenses and digital image captures can match the human eye.   In addition, our other senses color how we see things.  I imagine the air that you're breathing as you view the mountain sunrise has an effect on your mood and how you see.
On the other hand, an artist can take a digital photograph and using a number of techniques, capture the essence of the scene.  The edited photograph may not resemble the actual scene, but may evoke the same feelings.
A good example of this would be painters who paint on location.  The west has a number of great landscape painters who captured sunlight and mountain scenes.  On this side, we have Frederick Church and the Hudson River school of painters  who captured mere glimpses of daylight into almost immortal images.
if you have a camera with manual controls , try taking a number of pics with different exposures and shutter speeds to capture different views.  there may be one that matches what you see.  

I sympathize. I've never been happy with my mountain photos, often consigning 2 weeks of profligate photography to the trash without a single keeper. Even the photos by pros has seldom satisfied being preternaturally bright and focused. My view usually being filtered through a salty haze of sweat. The first and last light is hard to capture because it is so fleeting that it is impossible to capture the instant of maximum beauty which is only recognized in dim memory. My most disappointing photos have been taken in spring when the snow is so bright you can feel exposed flesh crisping and burning on forehead and cheekbones while you are squinting through blackened glasses at an incandescent world. At work at best you can show scenes in which the snow appears as bright as raw pork fat.

When photography burst to the scene the question of painting vs. photography was asked. The basic answer given was that painting captures the heart of things while photography the exterior. I'm not sure that's the best answer.

Philip mentions some painters trying to capture a specific light. Probably the most famous one is Monet (or for the British Turner) looking at a single scene in many different lights. For example, his Rouen Cathedral series of 17 paintings:



If painting sees to the heart, why do you still need the different lights?