Now you're right that they have more money and they can spend it. Some things like hiring a lawyer to sue someone are too expensive for an average person but accessible to billionaires. Rich people can do things with taxes loopholes that aren't practical for the average schmoe.
It is true that they often have power at their company and sometimes they use it overtly or covertly. But even this can be limited because they have to work with partners and other shareholders. The CEO of a big publicly traded company can't just break the rules because they're on a power trip.
Don't be obtuse. The people we're talking about don't go to the Post office, or the grocery store, and they certainly don't queue up. They don't even queue up at the airport, they have private planes, private security, private everything.
And "they only get one vote each election day". Voting is the least amount of political influence that a person can have.
There's this mythology built around great wealth and it's largely false. They can't just snap their fingers and make things happen as if by magic. There's no special magic power that only they get. They have to pay for what they want and just like normal humans, they can only spend the money once.
Just like how someone who has $200k can get a mortgage to buy a house, whereas someone with only $20k cannot. That's economic power that comes from having wealth to leverage.