Short answer: The difference lies in the transfer of wealth to the next generation and our meritocratic values.
As a society, we have a value system that we implicitly adhere to. At least in the US, meritocracy is big part of our value system.
We believe that if you work just as hard and contribute just as much as your neighbor, you deserve to enjoy the same quality of life.
So, let's look at some examples:
Example A) Person A works his/her ass off and becomes rich off the fruits of his labor. He then spends his riches on private jets and private health insurance.
Does this conflict with our meritocratic values? No. He deserves a higher quality of life because he worked his ass off.
Example B) Person A then purchases access to an elite university for his child, who as a direct result, enjoys a better life than his poorer peer who works just as hard.
Does this conflict with our meritocratic values? Yes. His child didn't deserve their higher quality of life because they didn't work their ass for it.
You don't have to agree with meritocratic values, but it explains the difference.