It is like the story of MP3 vs CD music, it depend on the ears that people have. For most people MP3 is the same, but some perceive the better quality of CD.

Not only the ears are important, the quality of the source is also important. If the music is a crap it will not make any difference between CD and MP3, even with 8 bit PCM it will be the same, crap is always a crap.

With jpeg compression is the same story, something is lost in the compression process. What is lost and how important is the loss will depend on the eyes and the quality of the source.

Many people don't see any difference between a 256 color gif and a 24 bit color image, I suppose that here everyone knows the difference. The same is jpeg vs bmp/png/tiff/psd/tga/etc. With training your eyes perceive more, more and more and something that once look the same now is not and the differences can make an image horrible that once it was good.

The question is what is lost in jpeg compression? Jpeg compression works in the wavelet or frequency domain where the high frecuency information is removed, the amount depend on the compression level that always exist some. Also the color information is reduced as the eye is less sensitive to color than to luminance, the same story as for color TV.

Here comes again the importance of the quality of the source, if the source has no or little high frecuency information nothing is lost with jpeg compression. A poor quality image has little high frecuency information, so nothing is lost. If the image has excellent quality and has a lot of high frecuency information the image can suffer a notable degradation in the compression.

Not all good quality images has high frecuency spectrum, an image that is a green plane has no frecuency information, well you also cannot define its quality. Repeated compression of the same image will reduce even more the high frecuencies, but as almost all was removed in the first compression very little remain to be removed in the second and almost nothing in the third. Compressing 100 times more will change nothing.

Now, what does mean high frecuency components, how you see them in the image. You see in how much detailed is the image, how much definition it has. A blurred image has only low frecuency components. A sunlight landscape full of trees, plants, blue sky, clouds can have a lot of of high frecuencies. The differences are much greater with digital cameras. A human face has little high frecuencies.

Jpeg compression has other big problem. The compression process divides the image in 8x8 pixel blocks, tranform each block to the feecuency domain, remove the high frecuencies and transform back to the image domain. The problem is with the 8x8 block, that is too small, a 16x16 pixels would be much better, but it would take a lot more time to transform. This 8x8 block has little problem with normal images, but with small images it becomes very important. A 640x640 image has 80x80 = 6400 blocks, but a 64x64 image has only 8x8 = 64 blocks. The result is that for very small images as thumbs jpeg cpmpression is horrible, any 256 color gif looks much better. And if you put some text in the thumb the result is even worst.

Conclussion. It all will depend on what you do. If you compress an image and see no difference or the difference has no importance then use jpeg because it will produce smaller files, even you can increase the compression level. If you are able to see the difference and it is important then save the image with a lossless format.

Another important point, even your eyes don't see any difference today, tomorrow maybe will see it, but tomorrow probably the cameras will be much better than today and so the stored images will have worst quality than the newer one, so it will not make any difference if are png or jpeg.

For professional use, never save images in jpeg format, something always is lost and you have to process and manipulate the images several times, you must preserve the quality n all the process even the final destiny is to be printed in a toillet paper with four colors.

Stupidity also evolves!