How did the different monastic orders develop such distinct styles?

Alright cool, something in my wheelhouse. Obligatory foreword: It's been a while since I did any kind of liturgical reading and I don't have the benefit of my old notes at the moment, but I'm going to attempt in broad strokes to answer your question. Further, my research is in later medieval western European religious practice, which means that there's a lot of monasticism that I can't really speak on. Why Ethiopian and Coptic monks practice differently is an interesting question but I won't be addressing it here. Finally, this answer is going to lean male, but there are really interesting examples of distinctive female monastic style, and I'll be discussing my favorite example of that as well.

It's also worth noting that "distinct styles" can mean a couple of different things. "Style" of course can refer generally to the way of life of monastics, or to their specific constitutions, or to their dress. I suspect that you might be asking about why monastics wear different habits, but I'm going to try to answer your question in a more comprehensive way.

That out of the way, let's talk about monasticism in the Latin world. The general trend in monastic histories is that as a response to the laxity of monastic institutions of the day, a founder (often a saintly one) founds a new community which returns to the original spirit of strict asceticism that is lacking in contemporary monasteries. There's often a bias in these accounts, as traditionally the histories of monastic orders were written by members of those orders, but it's certainly true that the history of western monasticism has been marked by periods of reform.

So for our purposes, it's safe to say that the classic reason for founding new monastic orders in the high middle ages was reform. Cluny, which at its height commanded daughter houses across Europe, was founded as a reform institution in the 10th century. The Cistercian order was another reform order, founded in the late 11th. Both these communities wanted to return to what they called a strict observance of the Rule of St. Benedict, the archetypical rule of monastic life in the Latin world, but of course, what it means to return to St. Benedict's vision of monasticism isn't obvious, and there were differences between the orders' interpretations of the Rule. In fact, the founder of the first Cistercian abbey was originally a Cluniac monk who was dissatisfied with Cluniacs' observance - this, when Cluny was meant to be the heart of the monastic reform. Cistercians' insistence on manual labor, for example, and their use of white habits instead of black, were ways that they communicated their purification of the monastic tradition.

White wasn't a feature of original Benedictine monasticism, but was instead traditionally worn by hermits, which brings us to another style of religious life - the eremetic life. Religious orders like the Carthusian Order had a different approach to monasticism than groups like the Cistercians, and attempted to combine the communal life of the monastery with the isolation of hermitage, with monks (both ordained and lay) living monstly in silence and solitude, except when joined for communal prayer - this is codified in the Carthusian rule, called the Statutes. The eremetic life was also practiced by the Carmelites, who had their origin in the 12th century, when Latin hermits living on Mt. Carmel asked to be given a rule, which would be the Rule of St. Albert. Like the Carthusians, they practiced hermitage, but unlike the Carthusians, when their order expanded to Europe they struggled to adapt their lifestyle to urban conditions. I don't have a lot to say about the eremetic life except to demonstrate that not all monastic developments in the middle ages had to do with strictly observing the Rule of St. Benedict - for Carthusians and Carmelites both, the Rule of St. Benedict had nothing to do with their attempts at reform.

Third I'd like to talk briefly about the mendicant orders, which represent probably the most dramatic shift in religious life in the later middle ages. While the Carmelites we just discussed were their own strange breed of mendicant, the movement is more archetypically represented by the Franciscan and Dominican orders, or more properly, the Order of Friars Minor and the Order of Preachers, respectively. To contemporary observers, mendicant friars and monks look pretty much alike - they both wear habits and so forth - but in the middle ages, the mendicant life was a stark departure from the assumptions of the religious orders. The word "mendicant" comes from the Latin mendico (to beg) and indeed, the original mendicants did not work (like we discussed Cistercians doing) but rather depended on alms for sustenance. We don't have the space here to discuss all the ways that mendicants were different from monks, but I do want to talk about a really fascinating book by Herbert Grundmann called Religious Movements in the Middle Ages (about to turn a century old but still cited by just about every book on medieval religion that I've read), in which Grundmann looks at medieval mendicants and medieval heresies and argues that they both represent responses to a common impulse to live the vita apostolica, or the apostolic life, and return to a supposedly purer form of Christianity from the past - put crudely, applying the Cluniac/Cistercian attitude towards the Rule of St. Benedict to the entire project of Christianity. Of course, Dominicans and Franciscans didn't share the fate of medieval heretics, but rather were able to integrate themselves into Catholic orthodoxy, and furthermore were often quite successful and powerful as urban institutions. Operating in the urban sphere, as opposed to monastic institutions' manorial organization, is one example of the ways that mendicants operated differently from traditional monastics.

This is getting long already. In a follow-up to this post I'd like to address the way that this response is gendered and speak a little bit to the experience of monastic women. First, though, while I realize that I've only scratched the surface of what's a very broad question, these are the general takeaways that I'd like to emphasize:

i. Reform was a common theme in medieval monastic developments, but reformers did not all agree on the interpretation of shared materials, nor did all monastics share a common rule.

ii. Monastic reforms and the mendicant movement arose from different circumstances, and faced diffferent sets of challenges.

iii. The climate of religious life (Mt. Carmel vs. Europe, the city vs. the rural monastery) affected the options available to religious.