80% of back pain gets better by itself, and a lot of pain may be from the paravertebral muscles (muscles around the spinal cord) spasming.
Make sure you have something that can be operated on first eg., demonstrated disc protrusion/nerve root impingement on MRI. Spinal surgery is not without its risks - paraplegia being the most serious one, although extremely rare. There is about 15-20% recurrence of disc protrusion. If you have private health insurance that helps a lot.
In the end, it really doesn't matter whether you go with a neurosurgeon or orthopaedic surgeon, as long as they are reputable. The best spinal surgeon in Melbourne is probably Peter Turner.
David de la Harpe is good also, as someone mentioned. Both are orthopaedic surgeons.
If you have nothing that can be fixed with conventional surgery, Professor Peter Teddy is the way to go. He is a neurosurgeon who specialises in chronic back pain. He can trial you on nerve stimulators eg., little devices on the skin, which help a lot of people with intractable, otherwise inoperable pain. If these work well, he can implant a permanent electrode into your spinal nerves to decrease the pain. You will need private health insurance to access these.
You can find all three surgeons by googling.