If you ask someone in HR, marketing, or finance who runs projects, many say IT does that. Then if you ask someone in IT who runs processes, they say that's what the other business departments do.
In Wikipedia, the definition of a "project" is "something frequently involving research or design, that is carefully planned to achieve a particular aim." It then says that projects can be further defined as "temporary rather than permanent". I guess the author just meant to say it has a start and an end.
Now let's look at how Wikipedia defines the word "process". There are many definitions, but the one we want is related to a "business process". The definition says it's "a collection of related, structured activities or tasks that produce a specific service or product."
Wiki Answers says, "A project is a unique endeavor with a beginning and an end undertaken to achieve a goal." "A process is a repetitive collection of interrelated tasks aimed at achieving a certain goal."
I would like to make the point that a process and a project are interrelated like kissing cousins.
1) Projects desire processes.
A project needs a process, where it desires a collection of related, structured activities since the objective of the work is to achieve a certain goal. For example, implementing your product at a new customer site would use a smart process that has all the right connected pieces of work to get the customer up and running fast and smoothly. Note, the word "smart" is defined as continuously seeking better ways with creativity and value.
The problem with project management tools is that they don't give you this impression since their approach is focusing on a task list, where each one can be it's own world. Projects fail because we spend too little time understanding that a project is driven by a process under the hood. Our project process can suck and kill us. Or it can blast away issues, risks, and do wonderful things. One is just a list. The other is passionate about delivering the right knowledge in real time and displaying visual relationships.
2) Processes are passionate about projects
A process is like a project, where it's an endeavor with a beginning and an end. It may have a shared goal but the deliverable is unique each time. For example, hiring a new employee has the outcome of a new individual joining the organization, but the process of hiring is reused over and over. Like projects, the hiring process has a start and an end.
The problem with business process management tools is that most are focused on transactional, automated processes. Gartner and Forrester say that 80% of our business process are unstructured, collaborative processes. This means that people get involved to collaborate and get things done, such as the hiring example. Don't think that if it takes only an hour to do from start to end, it can't be a project. Projects can come in all sizes. So, can hiring a person be a project? Sure. Can processing a customer bill from start to finish be a project? Sure. Can you view all of your new employee on-boardings as a project portfolio list? Sure.
In summary, projects and processes share similar characteristics and really need each other. A project with a great process will be more successful. A process with project like methods like planning, execution, and governance will be more successful. In both cases, imagination driven with discipline will make sparks fly. But don't tell your aunt.