Imagine that you and 14 others form a committee. Decisions are rendered through majority vote. Your vote will be important, but rarely decisive.
But imagine if you have a veto. Suddenly, your vote becomes crucial to each decision; you can halt action, even if you cannot initiate it.
Five states currently reserve the veto power to themselves. Expansion of the permanent membership of the Security Council is likely to occur soon, but the debate centers on (1) which states shall receive permanent membership; and (2) whether the new states will receive a veto power. (There are alternative possibilities as well--say, a veto that would only be effective if three nations supported it; a veto subject to overrule on a second vote, etc.--but it is unlikely that the current veto-holding members will yield the straightforward veto.)
As I've noted earlier:
The argument against extending the veto is that this would increase obstructionism in the Security Council.
But that assumes a one-play game. In a repeat-play game, an increased power of obstruction might, counterintuitively, increase the odds of cooperative outcomes. If the 5 original members faced new members with a similar power, any of the five original members might be reluctant to wield its veto, knowing that the new empowered members might retaliate through future obstruction on issues dear to it.
Any less would relegate the new entrants to an only marginally enhanced position within the Security Council. They would continue to be second class citizens.
See my earlier conjecture here.
Extending the veto would demonstrate respect for the ability of certain governments to contribute to a peaceful and productive international order.
Of course, other nations would remain second class, even if India/Japan/Brazil/South Africa were given the franchise.