The only real difference is whether the acts are official or not. So the President is allowed to order the military to assassinate a political rival, but not to pay a private hitman to do so.
It's a pretty ridiculous and indefensible stance from the majority opinion.
As an example, the fact he's in charge of the military doesn't make everything he asks them to do official because he is obliged to follow the constitution. There are arguments to make against what I just said. It's as clear as mud once you start going through concrete scenarios.
We have front-row seats to how empires decline and fade away. I guess those who missed the 20th century decline of the British Empire (culminating in Brexit) can study this one.
But the Supremes haven't said what they think an official act is; that's a matter for the court of first instance, according to the Supremes.
To me, in the UK, it looks like vandalism. There's no clear law on what constitutes an official presidential act, and until there is, the Pres is beyond the reach of the law.
I don't think these very senior lawyers are fools, I don't think they made a mistake. They've deliberately fucked-up the US legal system (even more than it's already a mess).
Read the actual ruling. They haven't fucked up anything. They haven't actually said much. Any time you read media coverage of a supreme court decision you've already made a mistake. You have to read the source material.