Example sentence:
God, she wanted to go home. She wanted this all to be over (with), but she needed to deliver the papers before she could.
How does the meaning change if "with" isn't added? To me, it has roughly the same meaning, but I'm not very sure.
The two mean essentially the same thing in your example, but I believe there is a slight difference in interpretation. There is also a more significant difference when they are used in different constructions.
She wanted this all to be over.
Here, she simply wants something to be done.
She wanted this all to be over with.
She still wants it done, but there's also the implication that it's something that she's personally enduring and wants to get through.
The use of the preposition with implies that one or more things are together in something. Here, she and the thing she wants to see done are "together."
For instance:
She wanted the riots she was watching on TV to be over.
She's an objective observer of the events.
She wanted the root canal she was having to be over with.
She's directly experiencing something (or is about to directly experience something) and she wants to have already got through it.
Sometimes, over with is exaggerated with the expression over and done with.
Also, it's more common when using over with or over and done with to use it in the middle part of a sentence rather than at the end. So:
She wanted to be over and done with the root canal.
It's important to note that putting just over in the middle of this sentence results in a totally different meaning:
She wanted to be over the root canal.
Here, she actually wanted to forget about the root canal—rather than wanting it to be finished.
So, in the sense of wanting to be finished, over and over with can be exchanged at the end of a sentence. But only over with can be used in the middle of a sentence to mean wanting something to be finished.
It would be understood by a native speaker to have the same meaning of being completed.
She wanted this all to be over...
She wanted this all to be over with...
