With a ticket in the currency pouch, slayer masters offer two different options when assigning a new task. Choosing the first option does not use a ticket; the second option does. If the dialogue is closed, the first task is assigned and no ticket is consumed.
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The use of a ticket allows for back-to-back tasks (e.g a player who had an Order of Ascension task as the first choice, completes the task and gets a new task with the ticket), even though back-to-back tasks were removed due to player complaints on hard/annoying tasks.
You may still receive the option to accept a Slayer Master's slayer challenge while attempting to use a Slayer VIP ticket. This does not consume the ticket.
During the 2016, 2017 Slayer Winter Weekends and the 2018 Slayer Summer Weekend, all new tasks acted as if the player had a Slayer VIP ticket in their inventory.