Off the top of my head, "to finish" has a meaning closer to "end", whereas "to complete" has a meaning closer to "fulfil". Here's a quick comparison:
"I finished my studies..." this tells me your studies simply ended. Perhaps you graduated, but it's just as possible that you gave up, quit or were kicked out of university.
"I completed my studies..." this is very clear. It means that you passed your exams and graduated.
If you're talking about work, use "completed" to mean you fulfilled all the required tasks.