Any time you view a "raw" image on any device, one of two things is happenning:

• The raw data in the file is being processed and interpreted by the application you are using to view the image. That application may be a simple photo viewer built into the device's firmware, or it may be a sophisticated photo editor such as Lightroom or Photoshop. There is no single "correct" interpretation of the data in a raw image file. Each application can interpret the raw data in the file differently. There is no "one" way to render the linear 12-14 bit monochromatic luminance values contained in a raw file in color on a 8-bit three color device. The raw data must be processed to be viewed.
• You are seeing a preview jpeg generated by the camera that took the shot. This preview image is appended to the file containing the raw image data, along with the metadata generated by the camera. Many devices will use this preview image when you open a "raw" photo.

Some applications display the preview image until they can render an image created by interpreting the raw data itself. Many applications have user selectable options that allow the user to select what is displayed when a raw image file is opened: the jpeg preview or one of many possible interpretations of the raw data using an automated routine or one of many selectable default processing profiles.

Related questions:
Why does my Lightroom/Photoshop preview change after loading?
RAW files store 3 colors per pixel, or only one?
Why is there a loss of quality from camera to computer screen
Why do RAW images look worse than JPEGs in editing programs?
While shooting in RAW, do you have to post-process it to make the picture look good?
Why do my photos look different in Photoshop/Lightroom vs Canon EOS utility/in camera?
Are paler raw images normal for a newer sensor with higher dynamic range?

I forgot to mention that the camera saves 1 JPEG and 1 NEF per image. NEF seems to be Nikons own raw "file format..?" (from reading your post it seems more like a datastream). Shouldn't the way to interpret a Nikon file be standardized by Nikon?

Nope. If you could only get one possible image from a raw file, it would defeat the entire purpose of being able to save raw files. JPEG images are intended to be the 'final' form for distribution and sharing. This normally includes some "punching up" of color, contrast, and sharpening.

Raw files, on the other hand, are a starting point. Many photographers prefer a more 'neutral' interpretation of a raw file be displayed on the back of the camera so that they can see something closer to what they've actually captured as opposed to what they intend the final version of the image to look like. If your Nikon camera is showing different previews for the JPEG and NEF versions of the same image, then it means the camera is creating two different jpeg previews, one for each version of the image. The JPEG preview for the JPEG image likely receives the same exact processing as the actual JPEG file does, other than being reduced in size. The JPEG preview for the NEF file is apparently receiving less aggressive processing (but it is still being processed a LOT!).

For more please see:
Are paler raw images normal for a newer sensor with higher dynamic range?
How to know correct exposure for RAW shooting when camera show JPEG Histogram
Why don't cameras show an "accurate" histogram?
How to make camera LCD show true RAW data in JPG preview and histogram?