Not in the context of the Constitution. "The people" makes reference to American Citizens. "Person" makes reference to legal persons.
An alternate, legal definition is "a corporation."
Lets break down that definition and dive deeper into the abstraction.
A law is a piece of paper with words written on it that some people agree on.
A corporation is created by filing articles of incorporation. Again, this is a piece of paper with words written on it.
So, in reality using law to define a corporation as a person you have a piece of paper saying a piece of paper is a human being.
Legally, marijuana and cocaine are "narcotics" but pharmacologically they are not.
The law defines personhood for biological entities too.
One minute before birth, it's not a legal person. One second after birth, it is a legal person.
This is nothing new.
If you don't like the law, convince enough people that the law should be changed.
They are virtually synonymous. People is a plural form of person. Most people don't speak legalese.