There's some similarities. The old new math was all about learning abstractions early. The new new math is all about approaching math problems from many different angles to build a sense of when you'd use certain techniques and why they work. The common thread is a lack of emphasis on rote application of algorithms without explanation of how they work. Arguably, that's less important in the era of pocket calculators and computers.

I dislike the lack of emphasis on rote memorization. IMO the correct approach should be 1) understand the concepts, then 2) practice until such concept becomes muscle memory, which will serve as the foundation for the next level.

This is the same approach that sports/martial art/artisans use (deliberate practice, etc). The concepts are tools, and being able to use those tools intuitively and instinctively is key to advancing to the next level.

Consider the standard long division algorithm. Would you rather students become fast and accurate in applying it or would you rather students have the skills to understand what's going on and derive it for themselves? Drilling will certainly help with the former, but it crowds out time that could be spent exploring concepts. There's a balance to be struck.

No, Common Core mandates what kids need to learn when, not how they're taught.