Agree is a verb that is often used in English. It means to have the same views, emotions, etc.; to harmonize in opinion or feeling (often followed by with): "I agree with you." Accord is much more rare in English usage. We almost always use it as a noun to refer to an agreement. "Both countries have an agreement." Accord is used in this idiomatic espression: "of one's own accord": meaning, without being asked or told; voluntarily: "We did the extra work of our own accord."

I can’t explain to you what the difference is, but I can explain what the similarity is. I have never myself used “accord” as a verb, but the way I understand it “agree with” and “accord with” can both mean “be consistent with”. So “a theory that accords with the known facts”=”a theory that agrees with the known facts”.