This relies on two things; one being the more important.

Firstly, the climate; while Darwin and Sydney do share similar environments, one (Sydney) is situated in the middle of a agricultural-ripe belt of Eastern seaboard greenry whilst Darwin is possibly the most inhabitable area of ground for hundreds of kilometres. It exists in a shell of un-arable land, while Sydney has hundreds of kilometres of it.

This may have not been obvious upon initial exploration, but certainly encouraged growth in later years. Also, in more modern history you'll find that the Katherine areas of Western Australia and the Northern Territory are nowadays being properly irrigated and converted into very rich farmland. So, historical precedents are being reversed, but the reasons for more population in the southern coast are also historical.

When Australia was discovered by European explorers in 1770 it was at Botany Bay that Thomas Cook made his landing. This is the northern Sydney area, and also where Cook made a recommendation to the British government to setup a penal colony. So already, we have the colonising government prejudiced for this land. Perhaps things may have turned differently if Darwin or Cairns have been home to this recommendation, but as previously explained I doubt that.

So in the next eighteen years all plans for the operation of an Australian colony were done on the presumption of Botany Bay as the prime area. And so it was when the First Fleet landed in 1788. From then onwards, Botany Bay and Port Jackson (the nucleus of modern-day Sydney) acted as the capital of New South Wales - the only existing Colony at that time. It was thus the heart of receiving anything from the UK; including migrants.

So, in short; Sydney was the hub of all colonial British activity and was also more attractive for farmers hence more population where the food is.

Perhaps a more attractive comparison is between the booms of Darwin and Melbourne, Sydney being a more obvious explanation. Both Darwin and Melbourne were founded within a few years of each, in the 1830s. By this time Sydney was a populous city of several hundred thousand. Both Darwin and Melbourne were free colonys - free of convicts. But by 1850 Melbourne was larger than Sydney, and by 1901 it was being seriously considered as the capital of the new Australian Commonwealth. Darwin, meanwhile, remained a backwater under the governance of the Brisbane-based Queensland State some thousands of kilometres away. This is because, in two words - gold rush. While Sydney was the hub of all British migration up until 1860 - Melbourne took over this role during the Gold Rush. Victoria housed some of the world's richest gold fields, and so the population trebled.

Darwin, basically, stood no chance. It had no arable land, Sydney was a inital migratory hub while Melbourne was the succesor migration hub for the remainder of the century. By 1901 the majority of the population called these two states home.

As a point of interest, all these explain this lack of population but I would point to arable land as to being the strongest point. For while South-East Australia may have the largest population, and the Eastern Seaboard being the more dominant - the West and Tasmania both have superiority in numbers over the NT, hence them being States and Darwin just a Territory. This is due to their lushness and wealth of mining materials. Darwin just can't win out, except perhaps in its location only.

I hope this all explains it from a proud Australian; as an Aussie another good answer would perhaps be, 'Because Darwin's a dump', but perhaps that isn't the most searching answer! (And untrue nowadays I hear.)

Why does the OP think Darwin & Sydney exist in similar climates?