Based on the Icelandic settlement at L'Anse aux Meadows, the first Europeans arrived in the Americas around 1000 AD. Australia was discovered in 1606 AD by Janszoon, 600 years later.

Because you only need to navigate on a straight line from Western Europe to reach the Americas, while Australia is on the opposite side of the world.

If we use Columbus's and Cook's voyages as a starting point (considering they're better known and both led to long-term colonization), the dates are 1492 AD and 1770 AD, which is still a 278 year difference.

I understand why you might want to use Columbus due to the diconnect with earlier Viking travels, but I don't see why you would mention Cook. Cook didn't discover Australia for Europeans. He already knew it was there when he sailed.

Based on geography, however, I struggle to understand this. The Americas are so far from the Old World. Australia, on the other hand, is extremely close to Indonesia and Southeast Asia, which were known to Europeans beforehand.

In Columbus's time, Western Europeans only knew about these in an extremely vague fashion and from third (or tenth) hand accounts. For instance, when Columbus sailed west, he wasn't just looking for China. He was also carrying a letter from the monarchs of Castile and Aragon to the Great Khan. The same Great Khan of the Yuan dynasty who had been expelled from China in 1368, almost 130 years before. They were unaware of that. This is just an example of how slow and incomplete the news that traveled across Eurasia were before the Age of Sail. If you were to ask an expert late 15th century cartographer like Toscanelli to draw Eurasia, he'd do little more than copy the Ptolemy world map from 1300 years earlier and slap a "Catay" (China) on the easternmost Asian coast, a big island called "Java" (or "Great Java") to the southeast, another big island called "Cipango" (Japan) to the east, and a bunch of random smaller unnamed islands just to show that there were other smaller islands there. It is irrelevant if East Asians knew these areas better and had more accurate maps of them, because Europeans didn't have access to those maps. They only knew from a trader who had been told by a trader who had been told by a trader... that China, Japan and Java were big trade hubs to the east of India. That's why Europeans called them collectively "The Indies" at the time.

Furthermore, Europeans aside, there was tons of trade happening in Southeast Asia involving Indians, Chinese, Japanese, etc... Why did none of them discover Australia either?

The big trade routes went through the straits of Malacca to Java, then around Borneo to the Philippines and the South China Sea, and from there to Japan. That's still thousands of miles to the west of Papua New Guinea or Australia. And yet we know that New Guinea and Australia coasts must have been visited or loosely explored by Indonesian traders at different points (for instance, dingos descend from SE Asian dogs that someone let loose in Australia 4000 years ago). However, they found no established cities and markets there, and whatever trade was, if any, was minimal or quickly abandoned. If the Indonesians were not bothered to establish themselves there, the Chinese, Indians and Japanese would not do it on their own thousands of years later.