The privately-run ones usually offer MUCH longer hours and they also offer lots of extra-curricular activites such as a LOTE (language other than English), a physical education program with a specialised teacher and equipment, Music classes with a specialised teacher and equipment, Library sessions at a library with a librarian, art classes with a specialised teacher that include more sophisticated activites like ceramics where the clay is fired in a kiln etc. The private places are also usually very big on documentation. Every significant piece of art work is collected and placed in a folio and presented to the parents at the end of each semester... most pieces of artwork is not just labelled but documented in other ways like included a quote from your child about what they have drawn etc. There are photos of your child's 3D artwork and photos of them making it etc.

ALOT of work goes into providing the extra things at private kinders. I used to be the art specialist teacher at one for 5 years. Along with the documentation of artwork each child has their own daily diary which the teachers fill in every day outlining what the child has achieved. There are also pre-reading component to their learning... each child's reading skill is assessed and supported. This is mainly in 4yo kinder but this is a very in depth process that i don't think the government kinders offer. This process involves a lot of 1-1 time between teachers and child.