How do monastic rules differ in each tradition? Theravada monks have to follow the 227 Patimokkha rules and the Vinaya Pitaka from the Pali Canon.

But how is it for monks from the other traditions like East Asian Mahayana, Zen and Vajrayana traditions?

I read online that Tibetan monks have dinner after 5pm, which is clearly not allowed by Theravada rules. They don't seem to have to do alms rounds, and they can take Tuesdays off.

In certain traditions, can monks receive raw cooking materials and cook food themselves? Can they grow their own food? Can they manage the finances of their monastery?

And why are they different? Do different traditions decide for e.g. that the rules should be made more relaxed, more strict or more adapted to their local circumstances or to modernity?



In the Maha-parinibbana Sutta, the Buddha did say:


  "The growth of the bhikkhus is to be expected, not their decline,
   ... so long as they appoint no new rules, and do not abolish
  the existing ones, but proceed in accordance with the code of training
  (Vinaya) laid down"


But the Buddha also said:


  "If it is desired, Ananda, the Sangha may, when I am gone, abolish the
  lesser and minor rules.
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.