In Australia we vote for local members who are usually representatives of a political party, and the elected candidates then become Members of Parliament (MPs) who choose their party leader, usually by a private party vote, unless the leadership is not being contested. This makes sense to me as it is the MPs who make up the party team, so they should be happy with their leader.
In Europe and the UK (where the parliamentary model is almost the same as Australia's: the Westminster system), some parties have direct voting for the leader. I guess the thinking is that a leader who is popular with the public will be a good leader, but the MPs might think another member would make a better leader, or be unwilling to follow the publicly elected leader causing internal conflict.
Something I think is odd is that it is the MPs who decide who the candidates will be, so it doesn't seem as democratic as it could be (to the party members), since the MPs have almost chosen their own leader anyway. MPs can also cause a leadership spill by voting for a motion of no confidence in their leader, bypassing the wishes of the party members.
What are the perceived benefits of this direct voting model?