The most likely reason is the relative brightness of your camera LCD and your computer
I wouldn't judge if the image is bright just because it looks bright in the LCD. I would
instead use the histogram - start by taking a well-exposed image where you have a histogram that indicates the image is not too dark and not too bright. I would turn off "Active D-Lighting" for this, so that the JPG (on which the histogram is based) exposure/contrast is as close to the RAW as possible.
Now review the image on your LCD. Does it look about right for a properly exposed image? If not, adjust the LCD brightness.
Now view the JPG image on your computer screen. It should also appear the right brightness. If not you need to calibrate your monitor, or at least adjust the brightness.
For the best comparison, review the LCD in the same light as the computer.
The important thing is to start with an image that you know is properly exposed, which is why I suggest using the histogram. Otherwise you don't know if your LCD is too bright, or your computer screen is too dark.
As far as RAW vs. JPG, the JPG will have adjustments made to it in-camera. Active D-Lighiting and other settings will affect the overall contrast. I don't think these should make the JPG significantly brighter overall.