Is it like 'finish with' is more specific and more grammatically correct? Or are there any differences between the two?
- It's time you finished your book.
- It's time you finished with your book.
Yes, there are subtle differences.
In (1) -- "finished your book" means "finished reading" your book. You could use this to urge someone to finish reading their book, but the emphasis is that they read the entire thing.
In (2) -- "finished with your book" means "finished using" your book. This could be used if you are urging someone to stop using the book, maybe so that someone else can use it. The emphasis is on them to stop using their book, no matter how much of it they've read.