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.
EX:
Lisa
repaired the roof (she repaired it herself)
Lisa
had 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