Exponent vs. Proponent

The OED defines an exponent as: One who sets forth in words, expounds, or interprets; in recent use occas. one who ‘interprets’ music, an executant. Also, that which serves to explain or interpret.

A proponent on the other hand is:

A person who puts forward or advocates a theory, proposal, or course of action; a propounder, a proposer. In later use also more generally: a person who is in favour of a proposal, a supporter.

The principal difference, therefore, seems to me to be that whilst an exponent 'sets forth and interprets', a proponent advocates. A proponent provides more opinionated support for a proposition than an exponent who merely explains. That is how I see the difference.

I don't believe there is much difference in terms of each word's connotation. However, proponent is more commonly used.

Exponent is rooted in the infinitive expose, while proponent is rooted in propose. If I were to dig into the nuances of the two words, I would use exponent in situations where the subject is choosing to emphasize something that most people would agree is factually correct, and I would use proponent when the subject is proposing something that isn't necessarily true from an objective standpoint, or that is more subjective by nature.