Why did everyone believe that if Sauron reclaimed the One Ring the world would succumb to him, just like that?
Didn't he already fail twice in attempts at world domination, and once even while wearing the One Ring?
Elrond and Boromir (at the Council of Elrond) lay out a few good reasons why it's imperative that Sauron's ring is destroyed now;
His Ring was lost but not unmade. The Dark Tower was broken, but its foundations were not removed; for they were made with the power of the Ring, and while it remains they will endure.
Many Elves and many mighty Men, and many of their friends. had perished in the war. Anárion was slain, and Isildur was slain; and Gil-galad and Elendil were no more. Never again shall there be any such league of Elves and Men; for Men multiply and the Firstborn decrease, and the two kindreds are estranged. And ever since that day the race of Númenor has decayed, and the span of their years has lessened.
We were outnumbered, for Mordor has allied itself with the Easterlings and the cruel Haradrim; but it was not by numbers that we were defeated. A power was there that we have not felt before.
Some said that it could be seen, like a great black horseman, a dark shadow under the moon. Wherever he came a madness filled our foes, but fear fell on our boldest, so that horse and man gave way and fled. Only a remnant of our eastern force came back, destroying the last bridge that still stood amid the ruins of Osgiliath.
There's a few reasons why they might be terrified of Sauron getting his Ring back, even though they managed to defeat him the first time. Mostly, it's because they barely managed to win the previous war, and were nowhere near as prepared for another of its caliber. This time around, it's unlikely that they'd be able to defeat a Sauron who had learned from his previous errors and was prepared to correct them.
First, it's important to note that the movie takes some serious liberty here. In the movie, we see the victory over Sauron as almost a fluke: Isildur just happens to get close enough to Sauron's Ring finger to cut it off, effectively killing him. In the source novels, though, it's never really shown how Gil-Galad and Elendil manage to defeat Sauron, but Isildur only cuts off the ring once Sauron is down.
Having said that, there's several things that are different now than during the War of the Last Alliance:
The Last Alliance Can't Exist
The end of the war saw the death of many of the great leaders of both men and Elves, including Gil-Galad, the King of the Noldor Elves. It's generally accepted that Gil-Galad's strength was the only reason the Last Alliance managed to defeat Sauron; but Sauron permanently killed him in the battle. The Noldor Elves were practically extinct by the time of the War of the Ring, with only the population of Rivendell remaining. Elrond was a powerful elf, but he wasn't on the same level as the old High King of the Noldor. Similarly, the other races of Elves had begun leaving Middle Earth, seeing that their time was ending and humanity was taking over.
Similarly, the armies of Men had fractured after Elendil and Isildur's deaths. They were no longer the unified fighting force they had been under the Numenorian kings. When push came to shove, Gondor refused to ask Rohan for aid, and Rohan almost didn't bother to answer.
Lastly, the Dwarves were in pretty bad shape: Durin's clan had been decimated by both Smaug and the Balrog, and were in no real shape to muster an army. (We don't know much about the other 6 clans of dwarves, except that two of them didn't survive the First Age and none of them appear in any of the published works.)
Sauron Has More Resources
In conjunction with the previous bullet point, Sauron now had the ability to field way more soldiers than the Elves and Men. In the Last Alliance, there were hundreds of thousands of Orcs against about the same number of Men, Elves and Dwarves. In the War of the Ring, Sauron fielded almost twice that, to the Men and Elves' less than 125,000.
In addition, by the time of the Council of Elrond, it was known than Sauron now had one of the Istari on his side, while the "good guys" only had Gandalf.
Sauron Knew Better
It's strongly implied that Sauron's defeat only happened because he came out to face Elendil and Gil-Galad himself. Though the Last Alliance had beaten his armies a number of times, the casualties were heavy. The siege of Mordor lasted for seven years, and it seems (from what we know) that Mordor could be largely self-sufficient if needed. Sauron's decision to send out his forces to break the siege, in person, ultimately led to his defeat. In the War of the Ring, Sauron himself never participates in the fighting. Mostly this is because he isn't at full strength yet without the Ring, but there's no reason to think he would make that same mistake a second time.
In conclusion: The Last Alliance of Elves and Men (and Dwarves and Eagles and...) barely managed to fend off Sauron at the height of their respective military powers. Middle Earth was in no condition to do that again if Sauron returned to his full strength.