They mean virtually the same thing but I would probably use them slightly differently... with finish my assignment for a routine task and assignment finished for a little more force or emotion.
I'd love to go but I want to finish my assignment.
I want to finish my assignment before I have lunch.
I want to get my assignment finished. (I want it done, over with, put behind me... so I can do something else.) In real life, if I disliked the assignment for some reason, I wouldn't take possession of it by using "my." I would say I want to get this assignment finished.
İn Cambridege Grammer İn Use book, the definition about the subject is;
We use ''get something done'' to say that we arrange for somebody else to do something for us.
the roof (she repaired it herself)
the roof repaired
(She arranged for somebody else to repair it)
but in your example sentence the meaning is different.
I want to get my assignment finished(
I will do it myself , I will not arrange somebody else)
but this sentence can have two different meanings , this situation is ambigious in my opinion
it can mean ''I will arrange somebody to finish it( according to cambridge)'' or it can mean ( I will do it , I want it done , put behind)
What is your opinion about this situation?
Thanks in advance