So I sent a thank you email to a professor who wrote me a reference letter. I updated her on the results of my applications. She replied back in a really nice email congratulating me and giving me some suggestions for my future career. I feel kind of rude ending our conversation this way, but at the same time, I also think it might be annoying for her to see another thank you email from me. If I do reply, I will most likely just say thank you again, this time for your kind words, and so on, which sound a bit repetitive. What would be an appropriate thing to do in this situation?
I would e-mail the professor back thanking her again for her kind words and telling her how you would use her career advice because it is a bit rude to get a reference letter and then cut off the conversation.
While it may be a bit repetitive, remember that you can always add how well the suggestions that she gave you worked out. Also, a thank-you letter does not always have to be long.
Your analysis seems correct. If she didn't ask you any questions or imply the desire for more information, I think you can and should leave it at that, for now. Once a "sufficient" back and forth chain of thank you notes has occurred there is no reason to extend it.
However, if the need arises in future to communicate with her you can open with thanks for past help and encouragement. You could also, then, let her know of any success generated by yer suggestions.
As you note, some emails are just noise. Some are nice, but not essential. Some are essential. This sounds like it would be just noise and she is likely busy encouraging the next person.
However, if you do take her suggestions and things work out well for you as a consequence, you can then write with the appropriate thanks and update.
I seldom reply to emails such as the one you describe unless I have something to say that will benefit the receiver in some way. There are exceptions, of course.