It's not equivalent, but it's roughly analogous from the Java programmer's perspective. You would spawn a new process much like you would start a new thread, but the similarities end there.
"Whenever you would spawn a new thread in Java, you would spawn a process." While that is true, it represents a subset of cases where you would spawn a process.
Processes are meant to model the real world. I'd say the truth is closer to "Any time you would instantiate an object in Java, you spawn a process." That's not always going to be true, but more true than you might expect.
It is really something different. Depends on what axes you are comparing. It is analogous in one aspect -- a unit of execution. But is very different in that it doesn't share memory. That second part is crucial and it what makes it have better reliability, better GC characteristics and so on.
