"I tried to look for ways to extend, but you can’t compromise in a high performance environment, nor did I want to compromise my family,” the 51-year-old, who was contracted until November, said in a statement.
"I am confident that there are many good people around to ensure the continued success of swimming in this country, and that the team is ready to face any challenge."
The former coach of Dutch champions Pieter van den Hoogenband and Inge de Bruijn, Verhaeren was hired in the wake of Australia's flop at the 2012 London Olympics, where the nation claimed only one title despite bringing a slew of swimmers with world-leading times.
Australia reasserted its claim as a global swimming power under Verhaeren with seven titles at the 2015 world championships in Kazan, but again stumbled on the biggest stage, leaving the Rio Olympics with three golds from 10 medals.
Taylor, the former coach of Australia's triple Olympic champion Leisel Jones, will be tasked with converting the country's world class talent into Olympic gold at Tokyo.
"Jacco has laid the groundwork for our Olympic campaign and I have worked closely with him and our Olympic leadership team," Taylor said.
"So (I) believe I am well placed to see those plans through and lead our team to success in Tokyo next year." (Reporting by Ian Ransom; Editing by Peter Rutherford)