I understand that the terms Cum Laude and With Honors are interchangeable, but which one is better understood in US and more commonly used?
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Both forms are well understood by people who care about such matters. You are free to use either.
Still, it is the diploma-granting institution (college or university) that decides whether to use English or Latin. My own baccalaureate was awarded "with highest distinction" — not summa cum laude. If you can find out what words were on the actual diploma, use those. If not, it doesn't much matter.
There are some differences in usage. I have never heard a graduate degree being awarded With Honors. "Honors" is for me usually associated with a minimum grade point average (quantitative measure) and "[Summa] Cum Laude" being a discretionary, qualitative measure decided on a case-by-case basis, although it could be based on numbers as well. In this I agree with Dorian: