How do monastic rules differ in each tradition?

The different in practices came shortly after Buddha's Parinibbana. That's is the reason First Buddhist council was convened. Ref Link: First Buddhist Council

According to this record the incident which prompted the Elder Mahakassapa to call this meeting was his hearing a disparaging remark about the strict rule of life for monks. The monk Subhadda, who had ordained late in life, upon hearing that the Buddha had expired, voiced his resentment at having to abide by all the rules for monks laid down by the Buddha. Many monks lamented the passing of the Buddha and were deeply grieved but Subhadda spoke up to show happiness and relief that Buddha was gone.

And Subhadda, the late-received one, said to the Bhikkhus: "Enough, Sirs! Weep not, neither lament! We are well rid of the great Samana. We used to be annoyed by being told, 'This beseems you, this beseems you not.' But now we shall be able to do whatever we like; and what we do not like, that we shall not have to do."

Let's go back to the basic that every Buddhist knows. According to The Nine Special Qualities of the Buddha, there are 'The Six kinds of Speech'. Whether the speech is disagreeable to others or not, Buddha only speaks true AND beneficial. If you believe in it, Buddha mentioned 227 Patimokkha rules which are true and beneficial for monk. It is also beneficial for those who makes offerings to monks who live by those Patimokkha rules. Please look at Qualities of the Sangha also.

All Buddhist traditions including the now extinct ones derived their rules from the original Prātimokṣa (path to liberation).

A number of prātimokṣa codes are extant, including those contained in the Theravāda, Mahāsāṃghika, Mahīśāsaka, Dharmaguptaka, Sarvāstivāda and Mūlasarvāstivāda vinayas. Prātimokṣa texts may also circulate in separate prātimokṣa sūtras, which are extracts from their respective vinayas.

The Wikipedia article on the Prātimokṣa covers the current and extinct derivatives in a little detail.

The monastic rules aren't inviolable, though generally through the ages their importance has been largely respected, and only modified in exigent circumstances. Philosophical stances of the founders have often influenced how closely they stayed to the original.

For example,

The Dharmaguptaka sect are known to have rejected the authority of the Sarvāstivāda pratimokṣa rules on the grounds that the original teachings of the Buddha had been lost.

When the Buddha began teaching there were no rules, the rules were added on as the need arose. For example, the prohibition on alcohol began when there was an incident of drunkenness that reflected badly on the monks.

Given the organic growth of these rules, as times and cultures changed some might have felt the need to change the rules to keep the tradition living and not turn into an anachronism. As Buddhism spread to new cultures the original rules as they applied to the Gangetic plains of India may not have made sense. For example, in lands with snowy winters like Japan obviously the robes needed to be warmer and with more layers than in India.

One other example would be the leave to allow monks to cook for themselves when Buddhism had just entered China. There was no Chinese precedent of donating food to mendicants and beggars like in India, so in order to maintain the respect of the monks in the eyes of the community, cooking was allowed. This continues today in modern Zen monasteries, like Plum Village in France, for much the same reason.

Some other times it may have been corrupted transmission or interpretation as the religion traveled across hundreds of years, several thousand miles and cultures.

The Buddhist scholar Tripitakamala felt the overall goal of Buddhahood overrides concerns for monastic vows.

cite: Gray, David (2007). The Cakrasamvara Tantra. New York, NY: Columbia University. p. 124.

Buddhism continues to evolve even today. The spread of Dharma in the West has brought with it changes to meet the unique challenges of the West. For example many modern Zen roshis in the US have dispensed with their robes except on ceremonial occasions, and wear every day lay person clothes. Chogyam Trungpa, the Tibetan master was a firm believer in such pragmatic change when he set up Naropa University and wore suits to lectures.