This may seem like dumb wordplay, but while management may seem to be about hiring / firing, budgets, etc, _leadership_ is about _influence_.

Influence and leadership are force multipliers. If you can lead people well and influence decisions (be it as a people leader, or a technical leader, or both), your experience can be learnt from by more and more people.

There is no doubt that leadership involves doing 'management' more often than not. But it's just one aspect, it's not the end in itself.