The button on the left (Tasmanian Police by Stokes and Sons) shows the cypher of the so called ‘King’s Crown’, or more accurately a Tudor crown. This indicates the button was used during the reign of either Edward VII (1901-1910), George V (1910-1936), Edward VIII(1936) or George VI(1936-1952). The top forms an arc.
The New South Wales Police button (back marked Amor Sydney) bears a ‘Queens Crown’ or more accurately St Edward’s Crown. This version with the gently rounded ‘shoulders’ on top is used for Queen Elizabeth II since 1952. A version with more angular or pointy shoulders was was in the reign of Queen Victoria (1837-1901). An example from diggerhistory,com is shown below.
Other versions of crowns have been used on medals and uniforms, and for the Navy. They are one way to date your buttons. The back marks can also help, as button markers and tailors existed for certain periods. Check out the relevant pages in this blog for more on this.