There are hundreds if not thousands of mircro-organisms grow in our tanks. We concentrate on the ammonia and nitrite eating bacteria as lack of them can harm, if not kill, our fish. The other micro-organisms have their part to play too. They carry out myriad other functions besides removing ammonia and nitrite, and help to stabilise things we can't measure. A mature tank is one that has grown a full set of micro-organisms, not just the two filter bacteria.
It is thought by a lot of people that neons are particularly sensitive to fluctuations in things we can't measure, things that are stablised by the mirco-organisms, and therefore neons shouldn't be put in a tank until it is mature.