When should I use "finish" instead of "complete," and vice versa?

They are kind of synonyms but not really interchangeable.

To finish means simply bringing something (a task or activity) to an end, or simply stop doing it:

We finished eating our meal.
_{(In this case you wouldn't use complete: "We completed eating our meal." is not the preferred usage).}

To complete means finish making or doing, such as in:

He completed his Ph.D. in 1983.

But another meaning peculiar to "complete" is making something whole or perfect, or with the meaning of "filling a form"::

• He only needed one thing to complete his happiness.
• Please complete the attached forms.

While to finish means merely bringing something to an end, or stop doing that something, to complete has the acception of fulfilling something.

Lastly, I'll leave you with this I found on the net:

"When you marry the right person, you're complete. When you marry the wrong one, you're finished."

When you finished your life, it does not mean it is complete; when you have completed your life, it is finished!