Exposure correction is using a single image to do something like HDR, but not true HDR. The reason is that you do not have the same dynamic range as three images provides. A true HDR will utilize multiple images, and have much more detail than a single image...meaning you will be able to see many more objects from the shadows of a single image, and you will be able to recover objects that are in the blown out detail of a single image.

When you do bracketing, the camera takes 3 images. One is exposing the entire scene two stops less than the 'base' exposure, but this will be correct for many objects in the image, and incorrect for others. Next, it takes exposures at 0 (base) and +2 (2 stops over), where you are exposing different portions of the image "correctly", and different portions "incorrectly". (I am using "correctly"/"incorrectly" in this instance to mean that there is adequate detail, vs incorrectly, meaning there is no detail available). In this case, each image has detail from different portions of the image that will be missing in the other images. When you combine the images, you are generating an image that combines all the exposed areas, throwing out the incorrectly exposed ones, thus leveraging ALL the detail available in all 3 images.

When you create one image, you expose for either an average of the scene, or a small portion of the scene, depending on camera settings. In this case, there are areas that are exposed incorrectly, and provide NO detail. In a single image, there is NO detail to be recovered, even in RAW...you simply will not have the information in your image.

The end result will likely be very different, however sometimes a single image "HDR" can provide more detail than you would get otherwise, and may be the photo you wanted. Just remember, you can not replace detail that simply isn't in your image, so its best to capture the detail as best you can, which means 3, or more images to capture all available detail. Also note that true HDR works best with RAW images.