All image processing packages should make this easy. I'll show you how to do it in Mathematica, if you have access to this system. Mathematica is a programming language, but it's really easy to do these kinds of manipulations, so if you have access to it (e.g. through a university site license), I recommend you give it a go!
First, import the image:
img = Import["http://farm1.staticflickr.com/62/171463865_36ee36f70e.jpg"]
Recompress it using JPEG compression
img2 = ImportString@ExportString[img, "JPEG", "CompressionLevel" -> 0.35]

Now take the difference of pixel values, converting them first to floating point numbers to ensure that negative values are preserved.
diff = ImageSubtract[Image[img, "Real"], Image[img2, "Real"]]

Not much is visible on the difference image (the difference is tiny), and negative values are clipped to black. So let's rescale all values to fill the whole dynamic range (the minimum will be scaled to 0, the maximum to 1):
ImageAdjust[diff]

ImageDifference
gives the absolute difference of the two images, and produces no negative numbers. This is the operation you are more likely to find in image processing packages, especially GUI ones (Photoshop, GIMP).
ImageDifference[img, img2]

We can also take a single RGB channel, for example the red one, and visualize the positive and negative differences using 'opposing' colours:
ArrayPlot[0.5 + ImageData[First@ColorSeparate[diff, "Red"]],
ColorFunction -> "RedGreenSplit", ColorFunctionScaling -> False]

Here's the same thing, with the differences amplified 5 times. The JPEG artefacts are more recognizable now.
ArrayPlot[0.5 + 5 ImageData[First@ColorSeparate[diff, "Red"]],
ColorFunction -> "RedGreenSplit", ColorFunctionScaling -> False]

The advantage of using a programming language is that we can easily automate this and see how the difference changes for "compression levels" between 0.1 and 1.0:
Grid@Partition[Table[
ArrayPlot[
0.5 + ImageData[
First@ColorSeparate[
ImageSubtract[Image[img, "Real"],
Image[ImportString@
ExportString[img, "JPEG", "CompressionLevel" -> c],
"Real"]], "Red"]], ColorFunction -> "RedGreenSplit",
ColorFunctionScaling -> False],
{c, 0.1, 1, 0.1}
], 5]
