The Roman Empire has given speculative fiction authors plenty of fodder for alternate history tales. There are dozens of stories in which the fall of Rome is delayed or prevented, and technology advances at an alarming pace.
For example, there's Jacek Inglot's Quietus in which a Roman Empire that has rejected Christianity has steamships by the 8th century and goes to war with a Christianized Japan. Or James White's The Silent Stars Go By, where the plans for the Aeolipile survive the fall of Rome and result in a starship launch in the 15th century.
Another great example is Frederick Pohl's The Deadly Mission of Phineas Snodgrass. Conceived as a response to L. Sprague de Camp's classic Lest Darkness Fall, (making, I believe, its third appearance in this column) Pohl's story concerns a time-traveler who brings modern medicine to the Roman Empire, thus creating a population explosion and forcing a second time-traveler to relieve the overpopulation by going back and assassinating the titular Snodgrass.
Thomas Harlan's The Oath of Empire series concerns an alternate Roman Empire protected by magic since the time of Augustus. The world's magic is elemental in nature, and Rome — in addition to surviving past its original expiration date — creates its own peculiar culture and habits based around that magic.
Perhaps the most depressing possibility is that a world with an extended Roman Empire would end up as a world-spanning dystopia with no real advantages over our timeline. A prime recent example of this idea is Sophia McDougall's Rominitas trilogy. In this series, the Roman Empire controls almost the entire world and has still not abandoned slavery in the present day. The world of Robert Reed's Hexagons isn't explicitly dystopian — however, his modern Roman Empire seems to be largely in decline and to has fallen behind against the still relatively simple technology of China and Japan.
A final, less depressing, and much more exotic divergence comes from Scot MacKay's Orbis, which is set in a world where the entire world fell to an alien invasion force 2000 years ago. The Romans, canny thinkers that they were, managed to steal some of the alien ships and escape off planet. In the book's present, a rebellion arises on Earth and humanity needs the help of the Romans who created an intergalactic empire.