I do not live in either of the two cities you mention (I presume you are talking about towns in the state of North Carolina?) but I want to give you some general information.

If you have not already looked up schools in the towns, by clicking on the "Research and Compare" tab on the green menu bar near the top of this page, do so. Once you enter the level of school (elementary) and city, you will find names of specific schools in the area, and you can click on the blue highlighted names of those individual schools to get more details, read parent comments (if any have been made,) etc.

Now, even if you find a public school rating, you need to know that it only compares to other schools in that state, but every state in the U.S. can determine its own curriculum standards.

Unfortunately, most people will say that public schools in the southern portion of the U.S. are NOT as rigorous as schools in the Northeast, for example. In other words, a school in a southern state might be able to say "90% of 3rd graders pass the state math test," but their math test only covers addition and subtraction. A school in Massachusetts (considered among the most competitive in the U.S.) might only say "80% of our 3rd graders pass the state math test," which sounds lower, but their math test may cover multiplication and division. In other words, ratings don't tell you much if you're trying to compare school systems in one state versus another location.
Here's a link to a site that looked at how children did on individual state testing in math and reading, compared to how they performed on a nationally standardized test: www.time.com/nochild

I'm making another generalization here, but I would say that if you can afford private schools, you will find them to be more academically challenging than a neighboring public school, no matter what state in the U.S. you live in. I'm afraid I can't tell you if a private school in North Carolina will be more challenging than a public school in Massachusetts, but it will most certainly have higher standards than a public school in North Carolina. Of course, there are exceptions to every rule, and my personal opinion is that an individual teacher, working with your individual child, is what really matters (there can be "poor" teachers at "good" schools, just as there can be "great" teachers in "lousy" districts.)

I realize I've not directly answered your question, but I hope it gives you more information to make your decision.