Yes, the raw data absolutely allows for better end pictures when the scene has high dynamic range.
A 14 bit sensor captures intensity with a resolution of one part in 214 = 16384. You are partly right in that largely the same dark to light range is captured, and that ultimately you will display or see the picture with much more limited resolution. Let's say the final picture will be 8 bits/color/pixel, which is only 28 = 256 levels.
The point you are missing is that it is important to be able to choose the final 256 levels from a continuum and not be stuck with the 256 levels evenly distributed from the darkest to the brightest point in the scene. Very often you want the final output levels to be non-uniformly spaced, sometimes significantly so, when mapped back to the original captured data.
Humans don't perceive light intensity linearly; they perceive it logarithmically. This means to get what appears to you as a fixed brightness increment, you actually need a fixed multiple in true light intensity. For example a 1.1x step by going from 50 to 55 will look like about the same increment as going from 200 to 220. Conversely, working this backwards, a +1 step from 200 to 201 will be hardly noticable, but a +1 step from 10 to 11 is quite significant.
A scene with reasonably high dynamic range can easily contain 1000x ratio in brightness from the dimmest part you care to see detail in to the brightest part. However, common display means don't come anywhere near that. Even a good print might be 50:1, or 80:1 under ideal conditions. The same holds true for computer monitors, especially since most are viewed with significant ambient light, therefore limiting how black black can really be shown.
Therefore, to make a final picture viewable on a 50:1 medium that starts out with 1000:1 contrast, for example, there needs to be serious compression of dynamic range. To make this compression look natural and acceptable to human viewers, it has to be done in the human conceptual intensity space. As I mentioned above, the human conceptual intensity space is the logarithm of physical light intensity.
The point of all this is to explain why linearly spacing out the 256 final display intensity steps over the original captured intensity range doesn't work. If you follow all the math, the result is that the limited display levels are grabbed from the original bunched up at the dark end and spread out at the light end. You can't do this and not lose information if you start out with only 256 linearly spread out levels.
There are other post-processing effects and other reasons for wanting to chose the display levels non-linearly from the scene intensities. All these require more captured brightness resolution so that you still have differences and smooth jumps between the limited number of output levels.
We just had a question where someone posted a night scene (high dynamic range) and wanted to know where all the detail in the dark areas went. See my answer, which shows a great example of what happens when you try to perform non-linear brightness mapping starting with only the same 256 levels you ultimately want to display the result in. Basically, don't let this happen to you.