Guitar strings wear out over time and are prone to oxidation due to sweat. As a result, they change their tone usually sounding more "bright" when they are new and "dull" when aged.
There's been a question on this site that goes into what changes happen to the strings i.e. that they oxidate and grease starts to settle in: Why does the quality of the sound of guitar strings decay over time?
However, no one has gone into how grease and oxidation changes the vibration characteristics of a string. In particular, I'd assume that it changes the way that a string is able to produce overtones.
I've heard strings would change their stiffness, but haven't seen a scientific comparison between the stiffness properties of new and worn properties anywhere. Does oxidation change the stiffness of a string too? How does stiffness affect the overtones generated by the string?
Also, it'd be interesting to know why changing strings is so particularly relevant on the guitar but seems to be less relevant on e.g. the Piano (to be honest, I've never heard of a string change on a Piano).