This is opinionated I feel. But for me I think the relation between the 2 is actually a massive mistake made by a lot of companies. There are a ton of software-systems hybrid jobs out there, and I've worked both ends of the spectrum, and in between. DevOps is a buzzword and it's new and shiny, so everything that falls in that realm is instantly deemed DevOps. I would argue that the same description you might give for a Production Engineer would sound exactly like what most people think DevOps is, when in reality DevOps(beware of opinions) is providing tooling for development to leverage and deliver to infrastructure. You are developing the operations of software shipping to and running on a platform. This makes a very diverse role, and that's why I think DevOps is more so a position than a profession. I consider myself a Software Engineer, but I understand the systems side of things very well so I apply my profession to a specialty. While I think, and have seen, that applied inversely, people who used to be sysadmins with knowledge of development and development processes.
I think a production engineer is similar loosely in the realm of work that is being done, but ultimately DevOps delivers solutions to and shapes the way the Development team works. Where as a production engineer is likely a systems software engineer who writes software to support infrastructure. They are writing software services which run on infrastructure in support of other services and not like DevOps might write Infrastructure as Code which is usually driving some sort of systems automation(puppet, chef, terraform, etc).