Please correct me if I'm wrong, but this is not the first meeting of the eastern and western churches. This is the first meeting between the Pope and the Patriarch of the Russian Orthodox Church, which has existed for around 400 years.

Lyon was probably the best chance of the two reuniting but by then it was too late.

As back story, 1274 was shortly after the Latin rulers were expelled from Constantinople. IN 1204 the leaders of the Fourth Crusade diverted ever so slightly from their plans to capture Jerusalem from the Muslims to sacking Constantinople and making it the capital of their Eastern empire. There is a lot to that story but it's not the point of this question. They were finally forced out in 1261 by Emperor Michael VIII Palaeologus. But the Latins had completely trashed the place. Not only had they looted and stripped it of most of its worth but they completely dismantled the Greek bureaucracy that was in place, which was a disaster. It completely disrupted trade routes and tax collection and efficient administration of things like grain distribution. They faced serious famine even as their population dwindled and dwindled. The Latins also attempted to replace of the Orthodox church and made Roman Catholicism the official church. But they didn't completely get rid of it. This wasn't all sanctioned by the Pope but he also didn't do much about it.

So here is Michael Palaeologus just reconquering the jewel of the East. And what does he get? A looted city and territory, decimated population through out the empire, a disrupted bureaucracy, and little means to do much about it. Oh, and the whole reason for that crusade, the Islamic expansion, still surrounded him.

Palaeologus knew that he needed an effective union with the West to survive. For all of their difficulties they were his only true source for allies. And one of his greatest risks as enemies. As such he was adamant about reuniting the two churches (take the obstacle of religion out completely). The council of Lyon went very well. He appointed a Patriarch who would be a valiant fighter for the reunification, John Bekkos. The emperor violently suppressed any opposition. If anyone was going to bring about unification it would be him (unless you had a pope willing to become Orthodox, which obviously would never happen).

But it was just too late. First of all, many of the people, who had long considered the West heretics anyway, now despised the Latins and the Church they associated with them. Understandably so considering their disastrous conquest and rule. Second, there were differences in doctrine that may seem small to those outside but were too big for the clergy of the Orthodox Church to ever accept. One big one was the nature of the Holy Spirit (did it proceed through the father and the son, as the West believed, or through the father alone, as the East believed). But other issues as well.

Essentially, it was incredibly unpopular with both the people, who wanted their religion, and the clergy. I think a key thing to think about in all of this is it wasn't the Catholic Church that would be making many compromises in this union.

Michael Palaeologus's son Andronikos II renounced the union. Poor John Bekos was removed from office and spent some of the next years under house arrest. But he remained a valiant crusader for unification all his life. Writing passionately about it and getting some traction with the church hierarchy to reconsider (there were some councils on reunification) but never actually making them change their minds.

In 1439, although the Latin conquest was less of an open sore, most of these problems remained. The people didn't want to become Roman Catholic. Some would prefer to be conquered by Muslims than be forced to be Catholic (which was perhaps valid since, although it was reduced in status and not as well maintained, a lot more mosques being built than churches, it's not as thought the Turks eliminated the Greek Orthodox Church). But, also, by this point there was little chance of a union with the West doing any real good. The western leaders weren't interested in saving Constantinople no matter how many calls for crusade the Pope may give. It was really a last ditch effort that was just too late.

Edited to fix typos.

PlinytheHipster and The_77 have given great answers. I've studied the Council of Florence pretty extensively, so I would just add a bit more detail to what they've said already. The relationship of the Emperor and Patriarch in the Byzantine Empire was, officially, much closer than that of the Roman Pope to any European ruler. Partly because of this, the Emperor (who was present in person at the Council of Florence!) was able to exert quite a lot of pressure towards unification. The degree to which he actually did so is a matter of much debate. Nevertheless, the council dragged on for months before any actual negotiations started due to a lot of procedural arguments. Then, when negotiations did start, there was a protracted period of stalemate while the same issues (mostly again the "procession" of the Holy Spirit) were debated without progress. When an agreement was finally signed, allegedly under increasing pressure, the Latin church conceded virtually nothing and the Patriarch died almost immediately afterwards. Consternation ensued. After the Greek delegation returned home in disarray, as mentioned, some defended the agreement, while others claimed (essentially) that they only signed off because the accommodations and food provided them were lousy and they just wanted to go home. Furthermore, the Byzantine man in the street didn't really trust the church hierarchy that much, in part because of their close relationship to the emperor. Greek Orthodox monks had a lot more popular influence than the clergy, but the monks weren't included in the Florence negotiations, with predictable results. As the monks and the people saw it, the emperor and prelates had betrayed them, and they were having none of it. Part of the reason that we now have several different Orthodox churches is because Constantinople finally fell to the Ottomans a little over a decade after these events.

Lyon in 1274 was driven by Pope Gregory X, and Michael VIII accepted to avoid a planned crusade by Charles of Anjou. The union of churches made such a crusade impossible. Michael appointed a new patriarch but was left with schismatics within the Empire and rejection of the union outside in Trebizond, Epirus and Thessaly.

However the election of a new pope Martin IV in 1280 friendly to Charles led to Martin declaring a lack of genuine reunion due to the large Byzantine opposition and excommunicated Michael in the autumn. Michael died in 1282 and Andronicus II immediately repudiated the union of Lyon and reinstated the old patriarch.

As far as the Union of Florence, the Emperor and all but two of his seven hundred delegates agreed to it. On return to Constantinople, there was some rebellion amongst the delegates. The patriarchs of Alexandria, Antioch and Jerusalem all repudiated their delegates consent to the union. The patriarch eventually chosen for Constantinople faced rebellion from his priests and bishops.

However, this did result in the crusade of Varna, where Ladislas King of Hungary and Poland was killed. The Union lasted till the fall in 1453, whereby Mehmed the conqueror installed an anti-unionist patriarch for his new capital. Without such a mandate as Byzantium leading the Eastern church and forcing Union, it was resisted by the Eastern faithful and was not practical to follow.

Source - A History of Byzantine State and Society, Warren Treadgold.