Final answer:
Forking a new process creates an independent copy with its own memory space, whereas spawning a new thread creates a thread that shares the same memory space within an existing process.
Step-by-step explanation:
The key difference between forking a new process and spawning a new thread is that process forking creates a completely independent copy of the current process with its own memory space, while thread spawning creates a new thread within the existing process, which shares the same memory space as other threads of that process. This distinction is crucial for understanding how operating systems handle multiple tasks and the resources they require.