What is the difference between them in this phrase? And Which one is not correct or are they both correct?
These things happen. We are all human.
These things happen. We are all people.
They both are grammatically correct. The issue at hand is whether there is common understanding between the speaker and the hearer as to what is meant.
If it is generally understood that "people make mistakes" then it just means that they understand that you made a mistake.
If it is generally understood that "it's human nature to be imperfect" then it means that they didn't expect perfection from you.
In either case, it depends on the context and mutual understanding between the two parties.
Either will work, but the first is more common.
There are two idioms here: "People make mistakes", and "We're only human". Both mean that people will often err, and that one should expect it. The first sentence above is more consistent with the idioms, but most listeners will understand what you mean if you say the second.
"We're all human" - feels either neutral (the simple fact), or scientific (our taxonomic classification is human) or like we are focusing on our faults (only human, not more).
"We're all people" - feels like we are focusing on the dignity of each individual, or trying to include others as equals. (We're all people, and therefore deserve _______).