I used to think those two terms (solicitor and conveyancer) meant the same, but they really are quite different. The difference is in the services these professionals can and cannot provide, and also in what they charge for their work.
Every time a property changes hands the process is called conveyancing. The transfer from one home owner to another has to be done in a certain legal way, and you can either do it yourself using a DIY kit, or with the help of a solicitor or conveyancer. Most people use professional help, because a house is probably the most expensive purchase they will ever make, and it is very important there are no mistakes in transfer of the ownership. An additional consideration in favour of using a solicitor/conveyancer is that they hold professional indemnity insurance, which protects you from their mistakes (you will get compensated if their error resulted in financial loss to you).
But if you decide to hand conveyancing over to professionals, who do you go to, a solicitor or a conveyancer? Basically, the difference is that a conveyancer can only give you legal advice about property, whereas solicitor can advise you about property and many other legal matters, such as tax implications, inheritance, etc. NSW Fair Trading website explains the difference between solicitors and conveyancers really well on this page.
In terms of money that they charge for their services, conveyancers are more affordable ($500 – $1000) and solicitors are dearer ($500 – $1500). And, of course, the cheapest option available is the DIY conveyancing, really – about 90$ for a kit.