What is the purpose of applying to a family court for a temporary order preventing the minor’s exit when the remaining parent has to give authorisation anyhow before the child can leave the country? What sort of authorisation is required?

An order preventing a minor's exit is a stronger safeguard – the mechanism of the remaining parent's authorization provided for may be too weak or open to abuse by a manipulative parent. For example, if prior written agreement is required an ‘abducting parent’ could forge this. The mechanism may not require written agreement, but just say agreement in which case the ‘abducting parent’ could claim he/she had oral consent, and leave with the minor.
The whole point of a temporary order granted by the court is that it can be registered at all ports ,with the Israeli border police, according to the I.D. and passport details given for the minor, and thus minimize the risk of the minor leaving the country . There is always the risk of the minor leaving on a false passport, or an additional foreign one, the details of which are unknown to the remaining parent, or being smuggled across a land border. Barring this, however, a 'stop order' registered at the points of exit from Israel should prevent the minor from being able to leave.
Preventing Abduction