Although to storm at somebody and to rage at somebody have similar meaning, I was wondering if there is any difference between them in terms of usage in daily life? Could you give some example situations for their usage?
Neither word is particularly common in spoken speech when used this way, so it's tough to be very definitive. Rage is most often used as a noun and storm is usually used in a more literal context.
They could be used as you describe in a literary or eloquent context, and there is a difference between them.
Normally a person will storm around, indicating that they are moving around, possibly pacing, visibly angry but likely not saying anything. Or they may storm through the door in an attack-like manner (for example, a SWAT team). Usually storm is part of a motion.
To rage is to be angry, often to yell. One is expressing rage, and there is no motion implied. One also hears of raging against something ("rage against the machine"). Usually this is an angrier version of complaining or verbally fighting against. I suppose it could refer to physical fighting as well.
I somewhat doubt thay any answer to this question is going to be more than individual experience. But, in my experience, "storm" as a synonym for "rage" is literary or figurative rather than colloquial. I do not think I ever sat in a bar and heard someone use "storm" in a figurative sense. Of course, my wife says I simply do not hang out with the right sort of people. Maybe I should find a higher class of bar.
In my personal experience using both of these sayings, to storm at someone would generally be figurative action "He was storming up the rear end" another more commonly used but similar saying would be "He came charging at me".
A general consensus of raging is someone who is angry and lets fly at someone else "He was just standing there raging at him, presumably because he had just run over his beer".
So I guess if you where to use them both in a sentence it would be "She came storming up to me and started raging at me because I accidentally left the toilet seat up (again)" (I promise that example is not from personal experience)
Perhaps these examples will help, because they are common in verbal and literary usage:
“I had to break up with my last boyfriend, because when he got behind the wheel of a car he turned into a monster filled with road rage.”
“The Rebel troops stormed from the rear of the guard to defeat the British invaders.”