Putin definitely understands the disaster on Russian culture and society that was the suppression of the Orthodox Church. He also understands that a renewed Orthodox Church can be a strong element of state power (like the Czars before him). He was, after all, a trained communist politician, so that’s how he thinks.
For the same reasons, I doubt he cares about Christians elsewhere, unless they are useful as a PR prop (which in this case, they are). Putin’s real views are unknown.
Certainly though - Obama will do nothing about protecting Christians.
The difference between the Soviet Union and Putin’s Russia, is that the Soviet Union positioned itself as the leader of the World Left, whereas Putin sees himself as the leader of the World Right.
He cares, because caring about Christians in the Middle East is part of a Tsar's job description.