I'm not talking about playing football every weekend or helping out at a nursing home. A friend got a work placement with a surgery team (don't know the details) because her friend's mother worked there. They're both at an elite private girls school. That's the sort of advantage I refer to. It's not fair or equitable, but you can't deny it exists.
But the point is that you don't need to go to an "elite private school" for these opportunities. You need to seek out for them by yourself or I guess, an over-exaggerated resume will do. 
Of course there are opportunities to be successful for both private and public school students. The issue is that it is generally much harder for public school students to have the same opportunities as private school students. At the least, public school students have to work a lot harder to build the network of connections that a private school student will have. If you do attend a public school, particularly a substantially under funded one, and have managed to secure an attractive work placement, you are probably the exception rather than the rule.
Some very crude hypothetical situations to illustrate:
A loves music and plays the piano. He/she dreams to be a successful musician. He/she goes to an under funded school which does not have a music course and accordingly cannot support students who have an interest in music. None of A's friends in school have much interest in music, either because they have not been able to afford to pursue a hobby in music or for other personal reasons.
B goes to a private school where there is a strong focus on extra curricular activities. This involves a large music program involving orchestras, regular performances and strong support for student musicians. Many students in the school have an interest in music and the school has a reputation for producing successful musicians who go on to have a career in music.
Of course A could work very hard to try and pursue his/her music career. He may even be a break-through success and become a world famous pianist. The point is, however, that the cards are stacked against him/her and A must work doubly hard to have access to the same opportunities that B has.
Another example would be two students who want to do law. One student attends a private school where many students have lawyers and solicitors as parents. Consequently, this student can easily build connections in the legal industry and when it comes to seeking placements, he/she will be in a strong position to get the lucrative positions. Another student who attends a public school may have no friends with lawyers as parents. As a result he/she must work much harder to build the same connections that the other student had simply by being in a different school environment.
This is largely the 'point' of going to an elite private school. They have an environment where almost everyone is going to come from a successful family. As a result it is much easier to make friends with the right people. Public schools by nature simply cannot offer the same opportunities which raises issues of equality and fairness.