So, what I'm looking to remove would be the plane inside the small box from the big box. This is just an example, so these objects will be complex. I can't find a boolean that will do this for me in Blender. My friend who has Maya says Boolean Union using Normals instead of faces does exactly this, but with Blender's Union it drops the bottom of the small box (and the plane inside the small box). I would love to be able to do this programmatically as well and without intersecting faces as this will be something that will need to be done a lot in a step of a computational engineering test bed, but I'll try not to get too greedy here. Please let me know if I am unclear on anything.
For anyone trying to do the same thing as I am here is my hacky way of solving it. I first did a difference as quiliup suggested, and then did a union with the original small cube, and finally removed duplicates. If anyone knows a more concise way to do this though I'd be happy to approve your answer.
I dont understand if you want to connect both of those objects. If so then boolean works. Just add boolean on one obj and then use Union and select the other mesh.
then just hit apply and your mesh should be cut. But both of them will be combined into one mesh with bad topology.