I've noticed that when I export RAW pictures, they look absolutely fine in Lightroom before being exported but after they look "washed out". I mean that when I make my selection before exporting to lightroom, in the "preview" module my pictures are colorful and nice, but when they're exported they're faded and look like I haven't used any setting when I shot them. I don't have this problem with JPEG. How do I solve this ?
It sounds like you might have Lightroom set to display the preview image embedded in the raw file, rather than displaying an actual conversion of the raw image data contained in the file. The preview image is a jpeg created by the camera from the raw data using the settings active in the camera at the time. It is attached to the raw file before it is sent to your camera's memory card. Sometimes it is not obvious from your Lightroom settings when you are viewing the embedded preview image, but if you have any of Lightroom's 'Preview' module rendering settings set to a 'fast' option, rather than a 'quality' setting, when you are in the 'Preview' module you're probably looking at the jpeg preview rather than a view of Lightroom's default interpretation of the raw image data.
Most in-camera jpeg engines increase contrast, saturation, and add some sharpening into the mix. These things are applied to the jpeg preview image attached to the raw file. Depending on what camera you use to produce your raw files and what software you open them with on your computer, sometimes those in-camera settings are also applied to the raw file when it is displayed. But Lightroom doesn't apply most in-camera settings to raw files. Instead, it applies whatever settings have been selected as the default options in Lightroom.
Of course in either case, you are not actually viewing the RAW file on your screen; if you're not looking at the jpeg preview then you are almost certainly viewing an 8-bit conversion of that RAW file which is similar to an 8-bit jpeg.
How do I solve this ?
You have several choices. One is to change the settings in Lightroom's preview module to display an actual conversion of the raw image data. This could slow down how long it takes Lightroom to render images, sometimes significantly. If you find that you prefer the "punched up" versions of your images produced by the camera own raw conversion settings, then create a Lightroom preset that closely matches what your camera is doing and select that preset as the default option to open raw image files.
Ultimately, the main reason to save images as raw files is to allow you to more creative control over the conversion process from raw to a viewable image that your screen can display. You need to use the tools in the 'Develop' module of LR to take advantage of all that LR has to offer. If all you are going to do is import your images into the 'Preview' module and then export them from there as jpegs, you might as well just adjust the settings in your camera before the shot and save straight to jpeg.
If you are using a Canon camera and open the .cr2 files using Digital Photo Professional (DPP) the in-camera settings selected at the time the image was shot will be applied to the preview image on your screen. Most other manufacturer's in house software does the same thing. Most third party RAW conversion software, such as Lightroom or DxO Optics, do not apply all of the in camera settings. Some of them will allow you to build a custom profile to apply to each image as it is imported or opened.
For more about what you see when you "view" a raw file on your computer, please see:
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?
For more about how to replicate your camera's raw conversion algorithms in external raw converters, such as Lightroom, please see:
How do I start with in-camera JPEG settings in Lightroom?
How do I get Lightroom 4 to respect custom shooting profile on 5d Mk II?
How to automatically apply a Lightroom Preset based on appropriate (Canon) Picture Style on import
Match colors in Lightroom to other editing tools
Michael Clark's answer is most likely correct. Lightroom is showing embedded thumbnails or the cache is outdated. This behavior can occur with other software as well.
If the cache is up to date and Lightroom is showing the proper image, the problem may be associated with some export options.
The colorspace of the images you are exporting could be Adobe RGB or ProPhoto RGB instead of sRGB. Unless every input and output device is properly calibrated on your system, using a colorspace other than sRGB will result in a washed out appearance.
The images will look fine in Lightroom and other Adobe software because they handle the Adobe RGB colorspace appropriately, but most other software will not. Fix by either calibrating all your devices (difficult) or making sure all exports are in sRGB (easier).
You may be exporting into a linear colorspace. This is possible when exporting high-bit-depth images. Linear color spaces look dark and flat because human senses tend to be logarithmic. Fix by exporting into a logarithmic-color format or viewing with software that support displaying linear-color images.