As conjunctions, but and yet are interchangeable. One is often substituted for the other to avoid repetition, as in this sentence:
Many, many people here share these thoughts, yet nobody can say anything. But I’m saying something. [Chatoyant Crumbs]
This has the same meaning as,
Many, many people here share these thoughts, [but] nobody can say anything. [Yet] I’m saying something.
Using one or the other in both spots would also create the same meaning, but it might sound repetitive.
Both words also work as adverbs, and in their adverbial senses they are not interchangeable. Yet usually means up to this time, while the adverbial but usually means only. For example, but and yet are not interchangeable as used in these sentences:
The health care bill is but a tentative first step on the road to needed reform. [Columbia Tribune]
Football coach Dabo Swinney said he has not yet had a conversation with Clemson officials regarding a contract extension. [Charleston Post Courier]
The results of an investigation in Seattle are but the latest example of a troubling trend. [The Root]
The board has yet to rule on the March petition. [Bloomberg Business Week]