Why do scholars still disagree by a century or more on when the Buddha lived?

Because the Buddhist sources date him not by a calendar but from a famous later king. They count how many years passed between the Buddha's death and the coronation of the emperor Ashoka, and they give different answers. The Sri Lankan "long chronology" says he died 218 years before the coronation. The "short chronology", from Indian sources and their Chinese and Tibetan translations, says 100 years. Both give him a life of 80 years.

Then you need Ashoka's own date. The Sri Lankan chronicles put his coronation in 326 BCE, which gives the Buddha 624–544 BCE, the dates accepted in Sri Lanka and South-East Asia. But Ashoka's inscriptions name Greek rulers whose dates are better known, such as Antiochus II Theos and Ptolemy II Philadelphus, and this places his coronation around 268 BCE. With 218 years that gives 566–486 BCE; with 100 years it gives 448–368 BCE.

So the possible dates range over more than a century. Many early 20th-century historians used 563–483 BCE. More recently, some scholars propose a death around 410 BCE, midway between the two chronologies. At a 1988 symposium, most presenters gave dates within 20 years of 400 BCE for his death, but not all historians accept this compromise.