Especially on an active battlefield, there's a tremendous difference between the two. The shelling of Monte Cassino is an example of "target/no-target" status. Reducing that historic monastery to rubble was highly controversial at the time (and since), but the debate was whether to make it a target because German troops might use its walls for cover, or to make it a "no-target" because of its historical and artistic value. That's a static decision that can be made fairly high up and far away. (The troops who were under the walls would have voted 100% to take the whole place down.)
A discussion of whether to protect Cassino monastery from looting by Italian civilians would be an entirely different issue. Whether to protect a particular building MUST be left to commanders on the scene, because the battlefield is a highly volatile, rapidly changing situation. It's not a "cop-out", it's placing decisions in the hands of the people who have the information to make them.
As far as the oil ministry, it's been thoroughly thrashed out either here or on another thread - by somebody knowledgeable about the business - that the records in that ministry are vital to keeping the oil wells functioning as a valuable resource for the Iraqi people. So they are higher up the scale of importance than the museum.
That all said, it's a shame that the artifacts are apparently missing. It would also be nice if we had had the building under surveillance from the git-go to avoid all the controversy about whether looting actually took place while U.S. troops were in the area, or whether that was simply a cover-up story. But I don't think blame can be laid at the feet of the military - protection of non-essential property is way down their list of priorities.