In my experience, 아/어도 is a stronger or more emphatic form than -든지.
I would translate the sentences you gave into English, with the following nuances of difference:
저는 무슨 일이 있든지 약속을 지킵니다: Whatever happens, I keep my promise.
저는 무슨 일이 있어도 약속을 지킵니다. No matter what happens, I keep my promise.
The second sentence, at least to me, carries slightly more emphasis.
Ultimately though, these forms are rather interchangeable in the context you are asking about.
One side note that I will mention is the use of interrogative (question) words with these forms. Consider the sentence "It does not matter when we go." This sentence has a question word in it. Usually it is considered unnatural to use the 아/어도 form here. It it is more natural to say 우리가 언제 가든지 상관없어요. When your sentence contains a question word, I would avoid using 아/어도.