The answer seems obvious. An operations manager manages operations and a project manager manages projects. This led to a few answers being posted highlighting the project manager's capacity to manage in the face of uncertainty.
I think that is an insufficient (but not wrong) answer because all managers are more or less operating in a degree of uncertainty, and it's things like process maturity and industry factors that drive uncertainty much more than the project/operations divide.
Years ago I read something (either in a textbook or journal) that talked about the frequency and impact of decisions in the context of operations and projects. The article described how operations manages made many many small decisions about how to get things done during the course of a day. Their focus is spread across many many things at once. Processes, people, products and services are all being dealt with, each getting only a minute or two of thinking time and sometimes much less. Project managers on the other hand are much more focused on a particular outcome. They spend more time considering options and assessing the likely outcomes from particular actions.
This led me to a 2x2 grid to help me think about the issue. Here it is.
Now, what are the risks associated with these quadrants?
The overriding theory is that big decisions made infrequently in a dynamic industry lead you to potential doom, and that smaller lower impact decisions give you the opportunity to steer and correct as you go. The business theorists say experiment small and often.
But there is a cost to decision making. One cost is time and money; we have to invest our team's time and money into analysis and round up the stakeholders for consultation and formal decisions/consent. Another is decision making rights; my boss has more than me and my subordinates have less. Of course we are all deferring to each other's specialties, but I still have the authority to tell a programmer how to do his job, even though I don't really have a clue how to program.
There are many techniques available to us for managing decision making processes on projects. It's important to understand the details of the costs, risks and benefits before selecting your processes. That depends on where you are working more than the type of project you are working on.
So back to the original question. What is the difference between ops managers and project managers? Maybe it's focus, and maybe it's the need to be able to switch between small and many decisions to important and few decisions. Maybe this is a reflection of the need to operate in the engine room and talk to the skipper at the same time. And even that is contextual.
Dammit. The answer looks like another "it depends."