The title is pretty self explanatory, but the context is that a new project with a small team will need the work of a product manager and the work of doing the UX both to get done. The company outside the project has some product manager skills, but very little UX skills. So deciding between a product manager with UX skills and and a UX designer with product manager skills depends partly on where those lines are usually drawn.
I've been working as a PM for almost 15 years and have had the privilege of working with 20+ different UX designers during that time, and have seen many variations on the PM and design relationship. I've also faced the same problem you have of choosing between hiring a PM gneralist and UX specialist.
Here's my advice: find a part-time UX designer (can be a contractor) and hire a full-time PM who has "good enough" UX design skills and the inclination to follow advice from (and learn from) talented UX designers. Here's why:
Net-net, if you hire a full time designer it's likely they will soon be bored and unhappy doing the job you describe on a small team. A PM with UX skills, however, will be thrilled to be doing UX work, even if it's just 10% of the time, especially if you can pair them with a consultant who can help the PM solve really hard UX problems and define overall UX guidelines.
If your app needs to be stunningly beautiful, meaning you'll keep your designer busier than most small teams can, ignore my advice above and go with the design expert.
Both jobs focus on the product, however they focus on it from different perspectives.
The PM, deals with feature requests, product performances, market response to the product, has a say on the go-to-market approach, prioritise features and so on.
The UX designer, both gets inputs from the PM, as in, "We need to deliver this feature, but there is this problem to solve", as well as providing him output from the user research, interviews, usability testing.
Of course the answer of a problem (design of the solution) can only be as good as the definition of the problem (product requirements).
It's also true that there are a lot of professionals that do both, (actually that's an assumption, but is true for me) and they are probably better defined as UX Architects, but they still don't replace fully a PM.
I think this question relies heavily on the scale and scope of the project at hand. This is also related to the size of the team/company that is working the project.
Generally speaking a Product Manager concerns themselves with a broad set of aspects of the work. They can both specialize and generalize, and in the realm of online product management I see them contending with some of the following elements of a project, in no particular order:
One of the key areas of expertise a PM can bring to the group is the ability to balance out the efforts related to all of the above and other elements of the project.
Whereas a UX designer is more versed in the specific UX and UI design as described above. And not at all ignorant of the other factors listed.
This is all from my professional experience, and I know others have had variations on the above. Also apologies for anything that may be missing, as I'm am trying to provide some general guidance for the questioner.
Where size and scale of project fit in is related to how much of a need the project has for the above specializations. In many smaller companies and on smaller projects, many times the Product Manager can fulfill the needs of the UX and UI contributions.
Hope this helps and happy to clarify.
At Marketo the difference is that the product manager defines the problem (MRD) and the UX Designer solves the problem (PRD). Additionally, product managers have like 10 other jobs to do.
Product Manager typically has to answer to the company. So they are beholden to profits and losses. The UX designer is more beholden to the end user. Obviously, they are both beholden to both, but that's how I usually divy it up.
Product Manager: This site needs to X Y and Z for the company. UX Designer: Let's figure out the best way to let our customer perform X Y and Z.