What image file formats are the most appropriate to use?

TIFF is the best image format to use for storage. The reason that TIFF is recommended over JPEG is because JPEG is an inherently lossy compression technique. This means that whenever an image file is converted to JPEG format, some detail is lost. However, as you have noticed, the changes that occur are very subtle at high "quality" settings of JPEG compression. You say that you cannot "see" any difference: can I suggest that you try:
Open your two specimen files, JPEG and TIFF, side by side.
Blow up to maximum zoom the same area on both images. Select a portion of the image with a good range of colours: an edge of the object, for example. Select the Eyedropper tool (keystroke "I" for shortcut) and make sure the "Color" floating tool bar is open.
Right click on any part of either image and select "Point Sample" (or adjust this setting in Eyedropper options on the "Options" floating tool bar.
Now left click on a pixel in the TIFF image. In the "Color" tool bar you should see the colour value of the pixel you have selected. Note this value.
Left click on the same pixel in the JPEG image. Note the displayed colour value.

You should observe that there is a general slight difference in the colour value at any specific point in the image. Indeed, it is very difficult to "see" this difference with the eye, but I hope that this numerical demonstration will prove to you that the two images are not identical. The JPEG compression routine does not store the discreet value of each pixel in the image, it stores a mathematical function that is used to re-generate the colour values and this process will result in approximate values for many of the pixels in the image.

Note also that TIFF files can be stored with LZW compression enabled, reducing the size of the file dramatically. LZW compression does not result in any change to the values of any pixels in the image, so is suitable for archiving and preservation purposes.

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