When do you agree and when do you concur?

As dutch, its more an educated guess. In general, they can be used for the same communication of messages. But I suspect subtle differences for experts to note.

But to concur goes seems to go deeper of shared insights, opinion and planning. Its more then simply say yes.
To agree seems more you can live with the plan, the outcome and proposal. Its a rough positive acceptance.

But corrections or additions are welcome

From what I found, concur is more formal. It also indicates that the speaker came to the same conclusion on their own, whereas agree doesn’t necessarily indicate that.

Agree is used more in every day communication. I think of both words as having the same meaning. I would most likely use concur if I was writing and didn’t want to repeat the word agree many times. Always good to have some synonyms in your back pocket to avoid repetition.

I would agree that the two words are synonyms and that “concur” is often treated as more formal. If we wanted to differentiate them, however, we might say that agreement is for situations where one person is presenting an opinion for others to evaluate whereas concurrence is for situations where multiple people are presenting opinions simultaneously to see whether they are consistent with each other. For example, we agree with someone when they successfully convince us of something. But we concur with someone when we look at the same information as someone else and both reach the same conclusion.

To concur is to arrive at the same decision point, although perhaps via a different path.

To agree is to to arrive at the same conclusion following the same criteria, conclusions, or steps.

So while I do not agree with @JLeslie, i concur with how and when she uses concur.

“I agree with you but…..”
“I concur”
When you agree with someone, you are giving yourself a chance of disagreeing at a later time. With a concur, you are in absolute agreement with the other person. “I concur but….” is never a statement.