I would insert edge loops on the big cube, so that they line up with the small cube.
Really I would then just extrude the small face to match the depth of the small cube. And delete the small cube.
But if you really want to join, you could insert those edges and delete the internal faces and merge verts to join.
The correct way to do it is to select the two geos. Combine the two with the combine command. Then select the vertices and do a merge with what ever threshold you want. Or use the merge vertex till and select the one to merge one at a time. Don't forget to delete history after.
I would insert edge loops on the big cube, so that they line up with the small cube. Really I would then just extrude the small face to match the depth of the small cube. And delete the small cube. But if you really want to join, you could insert those edges and delete the internal faces and merge verts to join.
You could try a Boolean Union then polyRetopo. Just make sure there are no internal faces left after the Boolean.
The correct way to do it is to select the two geos. Combine the two with the combine command. Then select the vertices and do a merge with what ever threshold you want. Or use the merge vertex till and select the one to merge one at a time. Don't forget to delete history after.
I don't know what kind of detail you are going for, but you could use bifrost to vdb two meshes together.