It's because they are both in a way powerful and influential and they have their own images to protect. They probably have lies that they want to keep secret so they just shut their mouth otherwise why would they be scared to really go at it as you said?
Elon Musk tried to but soon retreated back. I mean you don't target a president of a country and expect no retaliation or consequences even if these things are private.
With such opportunities as Musk, you can even attack the president of the country. Such a quantity of money can turn the world upside down. Therefore, the situation is not developing in the best way for Trump personally. To be honest, I like this kind of fight.
Besides, Elon went further and decided to create his own party, and this is already a serious intention. He has a lot of fans and supporters who will follow him to the ends of the earth. He is a kind of messiah of our time.