|Original Photo, Brightness Adjusted, Exposure Adjusted (Using Photoshop)|
As you can see, the brightness adjustment does not change the lighting of the scene realistically. It's almost like adding white paint to the image to make it brighter, which makes it look washed out and unrealistic. In contrast, the Exposure adjustment looks much more realistic - almost as if you made the shutter speed slower.
Now, GIMP has suffered with the lack of this feature. A post on the Stack Exchange forum stated that "Gimp doesn't have an Exposure setting like that, which is kind of a lacking point" and recommended that the person use the Curves feature instead. Similarly, there are more complex methods like the use of "layer masks" or colour correction. I might teach you how to use those advanced techniques in a later post, but for now those methods are just plain time-consuming.
Well, GIMP does have a little-known equivalent to the Exposure tool. It's called Levels (Color → Levels). To increase the exposure, move the middle slider (highlighted in red) to the left, and vice versa to decrease it. If you find that the image looks a bit too white, move the left slider (highlighted in blue) to the left to darken the washed-out areas slightly. Experimentation is key.
After around 30 seconds, I created this image in GIMP. It's not designed to be an exact clone of the Photoshop version, but rather to show that you can change the exposure realistically in GIMP as well. You might find this a bit more time-consuming, but that's expected of a program that costs you nothing.
Practice Makes Perfect
Download a copy of my source file here and try to make the image lighter or darker using levels. If you want to learn how to use the Levels tool properly, I recommend that you visit this tutorial on reading a histogram, the graph that appears above the sliders.