When a periodic comet flies by the Sun enough times, it can become depleted of its volatile elements. Once it is depleted, what distinguishes it from an asteroid, other than records of past sightings showing that it used to have a tail when it got close to the Sun?
Extinct comets are a sub-category of asteroids.
Minor solar system bodies are classed as "Comets" if they show a coma, or "Asteroids" if they don't. A depleted comet that has never shown any coma would be categorised as an asteroid.
It is hard to determine if an asteroid is an extinct comet. Some may be distinguished from other asteroids by its spectrum. F and B type Asteroids with low albedo may be extinct comets; comets are very dark. Asteroids with eccentric orbits at a high inclination are also often suspected of being extinct comets.
There is another meaning of "depleted", it means that the comet (with a coma, and perhaps a tail) is low in one or more species of molecule, which suggests it formed in an atypical location, or has been modified by some process since its formation. For example, C/2016 R2 is rich in CO and N2, but depleted in HCN. (Which suggests that it formed in a region of space at ~50K, so much closer to the sun than the outer Oort cloud.
A depleted comet would presumably have formed in the cometary halo, while an asteroid would probably have formed in the asteroid belt.
Comets formed several times farther from the Sun than asteroids, and so had a different chemical composition, and never became stratified into layers with different compositions. So even a depleted comet is likely to have a different chemical composition than an asteroid of the same size which formed in the asteroid belt.
I note that the origin of many asteroids is from larger bodies which one existed in the asteroid belt, bodies large enough have liquid interiors from interior pressure, and thus to stratify with heavier materials at the core. Then those objects collided with other objects and were shattered into thousands or millions of smaller objects.
Any asteroid resulting from the collision and shattering of a large body in the asteroid belt is likely to have the composition of the particular layer of that larger body that it came from, a composition which might be very different from that of an asteroid that is a piece of a different layer.
Meteorites that fell to Earth have several different compositions, and astronomers have found evidence those compositions correspond to various types of asteroids in the asteroid belt.
So possibly in the future there might be a program to shoot super powerful laser beams at small solar system objects, vaporize small sections of their surfaces, and use spectroscopy to analyze the compositions of the gases created by the laser blasts even better than the composition of the asteroids can be analyzed from reflected light.
If there is a program to classify the compositions of thousands or millions of tiny asteroids to find which should be mined first, a side effect may be to identify which "asteroids" are really depleted comets.
No doubt a solar system astronomer could explain the methods used at the present time to try to distinguish depleted comets from asteroids.