I created a swf file for one my clients using a screen size of 1024px x 768px (4:3 ratio), now my client wants to re-use the video in a bigger screen (TV) that still uses a 4:3 ratio but the screen size is considerably bigger (I do not have the exact size yet). Do I need to recreate the flash file using images that fit the new screen size, or the fact that they both use a 4:3 aspect ratio means that the image quality will not suffer?
It depends on what your SWF contains. If it's a video, then it makes sense to have different sizes (such as HD vs SD). However, if it's an application or timeline animation, you can look at setting Stage scale mode.
Usually when you create apps you would make them fluid so that they can resize depending on where they are, either full screen or within a set width and height.