I unfortunately can not speak as to why the Armenians became so widespread, as I'm embarrassingly not well-read on the Armenians and their diaspora (I am working on resolving that though). However I can give some context as to why some other groups from the Caucasus did not become renown for trading.

First it should be noted that historically the Armenians were not confined to the Caucasus. The modern Republic of Armenia is just a small sliver of what could be called Greater Armenia, and until 1915 Armenians were widespread across Eastern Anatolia. The genocide and further population transfers of the 1920s reduced their presence to the modern borders, but for centuries, indeed millennia, Armenians were found across the region.

That said, let's look at some other groups from the Caucasus. I'll start with the Georgians. Historically Georgians were never very urban or a mercantile people. They were primarily rural and agricultural, and it was actually Armenians who dominated trade in Georgian lands. The capital Tbilisi is actually only recently a Georgian city for this reason, as for centuries it was populated and controlled by Armenians; as recently as the First World War it had a majority Armenian population, and Georgians only constituted a majority as late as 1970. And aside from some pilgrimages to Jerusalem and some monasteries in Greece and Turkey, Georgians never really expanded outside of their territory, so could never become great traders.

Now to Azerbaijan. The concept of an Azerbaijani identity is a relatively recent invention, which only really began around the First World War. Azerbaijanis again were mainly farmers and peasants, with Armenians again dominating their capital Baku, and (along with the Russians), running the oil trade when that was established in the 1870s. Again this marks a lack of opportunities afforded to them.

Other groups from the North Caucasus, like the Chechens, Dagestanis, and Circassians, all were much less urban, and more focused on fighting off invading forces to establish trade networks (note that Chechnya and Dagestan were only nominally incorporated into the Russian Empire in the 1860s, and one can argue that they never really were, and still aren't).

Of course this still fails to answer why the Armenians were more urbanised compared to their neighbours, and what factors led to that. While I am at a loss, I can provide some reading and suggestions on how to find answers:

Perhaps the best work on the subject would be Richard G. Hovannisian's The Armenian People from Ancient to Modern Times. This is a two-volume work that looks at the Armenians, written by probably the most reputable scholar on the subject (he also has a multi-volume series on the 1918-20 First Republic of Armenia, published over some 25 years that I really want to read). Volume I, subtitled The Dynastic Periods: From Antiquity to the Fourteenth Century is probably the best bet to get an idea of how the Armenian trade network was established, and how the Armenians urbanised so well.

For a more concise look at the Armenian people, I'd suggest The Armenians: From Kings and Priests to Merchants and Commissars by Razmik Panossian. It's not going to be as in depth a study, but it is a more focused look at the topic as it looks at the development of the Armenian identity, which is the crux of this question.