To put it simply: cost and the continued influence of the Roman Empire. Honey is much more expensive to produce than grapes and grain, and always has been. By no means does this mean it wasn’t a popular drink. Prehistoric remnants of mead have been found all over the world, including China and India, but I am less versed in eastern history so I can only help illustrate the European perspective.
A well-rounded understanding of mead in Europe requires an understanding of three separate civilisations, spanning two millennia. To start with we have to address the most obvious, but also the latest: the Norse. While, of course, mead was a part of the culture long before the Viking age of the 9th and 10th centuries AD, this time period is when Norse culture bled into Britain and, to a lesser extent, Central Europe. Despite sharing a common Germanic heritage, there was little cultural exchange between Norse people and the rest of Europe before the vikings (an anachronistic name that originally meant pirate) started raiding them. The drinks of choice for the Norse were beer and mead, with the cold climates of Scandinavia preventing grapes from being cultivated there. Due to similarities in the presence of a pagan pantheon, the Viking age often gets lumped in with the Greeks and Romans, despite sharing no cultural roots and being separated physically and temporally by almost 1000 years. At this point in European history, the alcoholic legacies of Europe were already firmly established, and the popularity of mead couldn’t disrupt the ubiquitous beer and the more refined wine.
The importance of mead is demonstrated through its constant mention throughout Norse mythology. However, it is also present in Greek mythology. In Dionysos: Archetypal Image of Indestructible Life (1996), an interesting etymological point is made: the Greek verb ‘to be drunk’ is μεθύω (pronounced meth-ú-ō), which originally derives from Indo-European words for honey, making it cousins with the modern word ‘mead’. It is also noted that the Orphic cult of Dionysus (who had a much more archaic focus in their religious belief system) payed reverence to mead as the precursor to wine, with Zeus intoxicating Kronos with mead to castrate him. The Orphic tradition was one of hedonism and ‘euphoria’ as a form of religious practice, and so this interest in mead is especially notable, given the religious significance they placed on wine. Interestingly, Orphism saw its roots around the 5th Century BC, the Golden Age of Greece, during which mead was a common presence at feasts and other celebrations. Here, the impact of the cost of honey is seen. The Greece of antiquity was a prosperous civilisation, with the Persian War setting up the Delian League, eventually evolving into the Athenian Empire, before the Peloponnesian War crippled this prosperity. It is only in this time period that something as expensive as mead could become a truly popular drink. Following this period of Greek history, mead was seen as an expensive representation of their more glorious history.
It’s time, then, to talk about wine. The key difference between the Norse and the much earlier civilisations of Greece and Rome is their climate. Greece and, even more so, Italy have a much better climate for grapes, and the prevalence of wine is easily explainable. In the same way that the flat land of Central Europe is perfect for grain cultivation (hence German beer culture), grapes could be grown both in lowlands and alpine regions, leading to a wide variety of flavour in the same drink, and lending itself perfectly to cultivation in Italy. Wine was extremely common and popular in Greece, but the span of Greek imperial history in both size and time meant that it did not earn the same prestige that mead, the drink of the gods and the golden age, did.
Finally, we come to the Roman Empire. The Romans famously elevated Roman culture above that of the barbarians they saw themselves as being surrounded by. Therefore, beer was seen as an unrefined, barbaric drink. While the Romans revered the Greeks as their intellectual precursors, the cultural connection to mead just wasn’t there, especially when their homegrown alcoholic produce, wine, already served as an elevated alternative to beer for much cheaper. As the Romans expanded into the Germanic regions, they retained their love for beer, while also gaining appreciation for wine, while mead was left by the wayside.
Greece had a love for mead, but was too far removed to have an impact on our drinking habits now, while the Vikings were too late to impact an already established drinking culture. The idea that mead has ‘become’ unpopular is a false proposition: aside from specific parts of history, it was never really placed to overthrow beer or wine as the alcoholic drink of choice. This has led to a complete lack of a large market for mead in modern times which, combined with the expense of producing it, has led to a lack of mead in the global stage of alcoholic beverages.
You may be interested in this past answer from /u/Daztur, or this past answer from /u/phasestep :)