In image processing, what is the difference or relationship between noise and artifact?
The difference is that noise may obscure features in an image, while artefacts appear to be features but are not.  If the 'problem' is structured, it is probably an artefact, whereas if it is random, it is probably noise (as a generalisation).

A computed tomography example:  noise will make the image look grainy, and make small differences in contrast difficult (or impossible) to identify.  A streak artefact on the other hand has structure to it, and looks like the patient has a region of low density where they actually don't.  Likewise with ring artefacts - they look like the patient has ring shaped structures within them.  Artefacts are generally caused by assumptions made in the development of the reconstruction algorithm (though not always).

It is similar to the difference between noise and interference in some other fields - interference is structured in some way, having some particular source, while noise is less predictable.

EDIT: Just found something in the glossary of "Computed Tomography" by Willi A. Kalender, 2011 (978-3-89578-317-3):


artifact: part of the contents of an image that does not have a counterpart in the physical object being imaged; in CT images an artifact can appear as false structures (e.g. aliasing artifact, beam hardening artifact, motion artifact, partial volume artifact) or as a corruption of the CT numbers.
noise: contributions to a measured signal due to random processes; in principle these contributions are of statistical nature and do not carry any information about the signal.


If that's of any use to you.