Question
Apart from ensuring the lawn looks neat and tidy, why is is good and what are the good purpose of mowing the lawn?

Answer
It is a general truth that most plants respond positively to pruning. Pruning stimulates new growth.
Most grasses are improved strains of pasture or grazing grasses. These were grazed regularly every 21 days or so, usually by cattle. To promote sturdy lush growth, sheep were allowed to follow the cattle every so often. Sheep graze closer to the ground and the grass would "tiller" more profusely after such a treatment. You could say that lawn grasses have evolved to be mowed regularly. Mulching the grass clippings at mowing time returns nutrients to the soil and less fertilizer is applied - a saving of time and money; and less load on the environment. Depending upon its design, a mower has a vacuuming effect on the lawn - pulversing dead material thereby leaving a lasting and favorable  visual effect.
It has been said that you should treat your lawn like your hair. Cut (mow) it, comb (rake or vacuum it) and shampoo (soap) it on different but regular intervals.