I'm getting ideas for my own timeline now! 
Me too... let's start a "world without Rome" fad!
Anyway, here are a few things (not an exhaustive list by any means) known in the Hellenistic world:
Military technology:
- Non-gunpowder[1] siege weapons that made the medieval trebuchets pale in comparison[2].
- There were repeating catapults, and repeating crossbows wouldn't be too much of a stretch.
Naval technology:
- There was, as mentioned the ability to sail the open seas, which was possible because they possessed 1) a coordinate system, ie a scientific theory of cartography 2) reliable and 3) a method to locate the ship with respect to the coordinate system
- It seems that there was a push towards building larger ships. The descriptions of some of them make me think of Zheng He's treasure ships.[3]
- Canal-digging was pretty advanced, as there was a canal linking the Mediterranean and the Red Sea.
- Ships had lead-plating to protect from barnacles, of which none of the British and Dutch ships had as late as the seventeenth century.
- Very advanced catoptrics, the ability to build lighthouses. Pharos in Alexandria would be the most famous example, but many others had started to built throughout the Mediterranean.
Water engineering (this area should not be underestimated):
- In aqueducts, pressure pipes (simply called "syphons") were used, which overcame depressions in the terrain.[4]
- The Archimedean screw, a tool for lifting water.
Alternative (from muscle power, that is) energy sources[5]:
- The water mill was known, and used throughout the whole Mediterranean. Horizontal ones to boot, which are more effective than vertical ones.
- Windmills were in use, and so wide-spread that there were quite a few place-names named after them (anemourion)[6]
- The possibilites of steam power had started to be explored[7].
Intricate machinery:
- The Antikythera mechanism was found on a shipwreck outside the islet of Antikythera, between Peloponnesus and Crete. It was a sort of perpetual calendar that allowed the calculation of the phases of the moon, past and future. Two features stand out: 1) It uses at least thirty gears, which makes it almost seem like clockwork. 2) "[...] the presence of a differential turntable, a mechanism that allows the addition or subtraction of angular velocities. The differential was used to compute the synodic lunar cycle (moon phase cycle), by subtracting the effects of the sun's movement from those of the sidereal lunar movement".
Medicine, biology, botany, zoology etc:
- Anatomical knowledge was quite developed[8].
- Diagnosis, pathology etc. had been developed.
- There was measurement of the pulse.
- Mental illnesses had started to become explored.
- Biological classification was in full swing[9], also fuelled by the conquests of Alexander the Great, who himself ensured that flora and fauna was sent back for study.
- Fossils were widely studied, and many were identified as being of species no longer extant.
- There had been developments towards an evolutionary theory[10].
Chemistry:
- The problems with understanding what rate of progress Hellenistic chemistry was at is that it later morphed into alchemy - "a syncretism of Greek natural philosophy, Egyptian magic, allusions to Judaism and Christianity, craftsmen's recipes and empirical chemistry".
- What we can be quite certain of is that the artficial pigment industries, cosmetics and fragrance industries were quite developed.
- The conception of a molecule had forerunner in the oncos.
Art, music etc.:
- There is the possibility of primitive motion pictures[11].
- Figurative art was pretty advanced, with there starting to be more emphasis on painting rather than sculpture. An example of the new figurative art: http://www.ancientsculpturegallery.com/images/alexandria_HuntingDoggilded.jpg
- The novel.[12]
- The first keyboard instrument: the Ctesibius water organ.
- More advanced music started to develop.
- Greater interest in preserving cultural heritage, with traditional Greek songs etc. started to be written down.
- The birth of Greek grammar.
Some advances in agriculture:
- Plants from outside Hellenistic kingdoms started to be cultivated, and preexisting plants were improved through seeds imported from different countries.
- "Animals from elsewhere were acclimated, breeds were improved through crosses, and wild animals such as hares, dormice and boars began being raised, as did fish species".
- Egg incubators.[13]
- There were animal-powdered automatic harvesters with teeth and blades. Very simple, but beyond the ken of medieval and early modern Europe.
- Egypt's population around 1 B.C. was eight million, with a half million in Alexandria, and they were major exporters of grain. An estimate of Egypt's agricultural capacity in 1836 had it that eight million was the maximum population that could be fed if all land capable was cultivated...
- The production of olive oil throughout North Africa was very advanced, aided by the invention of the screw press.
Some advances regarding metals:
- There were drainage installations in mines, from Andalusia to Afghanistan.
- In early Hellenistic times iron came into common use for tools and machinery of every kind.
- "From the little information we have about metallurgical procedures we can glean certain technological innovations in the area of metal refining. Polybius tells us about a new blacksmith's bellow, perhaps fed by the Ctesibian pump".
- The clearest example of advances in metallurgy would be the Colossus on Rhodes; when in the Renaissance they wanted to build a similar structure, they had no idea how to go about it.
Lucio Russo also has written a word of caution, useful to alternate history writers:
I think there can be no doubt about the importance that ancient science and Hellenistic technology could potentially have had for production processes, but in assessing the extent of applications actually deployed in Antiquity we must avoid certain traps that lurk in making comparisons, whether explicit or implicit, with our own age.
In Chaplin's movie Modern Times, the tokens of modernity are screws, gears, transmission belts, valves, steam engines, automata: a smorgasbord of inventions from ancient Alexandria. How can one say that these innovations were useless back then? Yet, though so much of the technology that made up the movie's factory goes back to the third century B.C., it is clear that in that century there were no factories like Chaplin's.
The Western world has experienced since the late seventeenth century a unique phenomenon in human history, characterized by an exponential increase in several technological and economical indicators, and the source of achievements and problems without parallel. (This growth certainly cannot continue for long at the same exponential pace.) The primitivists are right in warning us against the pitfalls of "modernizing" Antiquity by reading into it the accoutrements of modern life. There was certainly no Hellenistic Industrial Revolution, there were no stock brokers in Alexandria and the Mouseion was not the Royal Society. On the other hand, using today's Western world as a sort of universal standard, lumping all ages other than ours into an undifferentiated "underdeveloped" category, can be highly misleading. If we think that biology has predetermined a unique possible path for the human race, culminating in the "economic rationalism" of today, it may be possible to define other civilizations by how far they are from ours; but human history is much more complex than that.
The application of scientific technology to production does not necessarily mark the beginning of the process in which we find ourselves now, where technology itself grows exponentially. Having made this clear, I think it must be agreed that scientific technology did have in Antiquity important applications to production. The Mouseion's economic role was not comparable to that of the Royal Society, but that does not mean this role was insignificant, nor does it imply a lack of wisdom or foresight on part of the ancient scientists. The process of exponential development starting with what is usually called the Industrial Revolution as triggered by a plethora of economic, social, political, cultural and demographic factors that we have not yet understood in depth. It is more sensible to try to figure out what happened in Europe in the late seventeenth century than to ask why the same thing did not happen two thousand years earlier. Hellenistic scientific development was violently arrested by the Roman conquest. We may wish to speculate on what might have happened had this interruption not taken place. Nothing authorizes us to conclude that things would have gone the way it did in seventeenth century Europe; we do know, however, that the recovery of ancient knowledge and technology played a major role in the modern scientific take-off.
[1] "The introduction of firearms in the modern age concerned primarily large-bore guns used against fixed positions; as a personal weapon, the arquebus took centuries to supplant the pike. So the role of gunpowder was to replace the catapult, the technology of which had been lost".
[2] Fortification overall did change as well, because walls started to become "thicker and started being surrounded by moats, but were complemented by towers capable of hosting catapults". The advances in siege outpaced advances in defense, though, as shown by a rapidly increasing amount of victorious sieges.
[3] "Merchantmen also got bigger. Hiero II of Syracuse had a cargo ship built, the
Syracusia [...] Thus we know that the ship, whose construction had required as much wood as sixty quadriremes, had on board, among other things, a gymnasium, a library, hanging gardens and twenty horse-stalls."
[4] "The most remarkable syphon was at Pergamum; it pushed water uphill to a height of perhaps 190 meters from the deepest point, and the pressure at the bottom must have been almost 20 atmospheres."
[5] Whoever holds Iberia is in a good position, as both wind and water energy is plentiful there, and there's even coal in the north.
[6] "Many scholars have felt that the Heronian passage can be disregarded because it is not confirmed by other writings. Heron presumably meant anemourion in a moment of distraction, forgetting that it had not been invented yet. We know that he was given to such lapses."
[7] "The first steam engine actually built in modern times seems to have been the one described in 1615 by Salomon de Caus; it operated an ornamental fountain intermittently. Thus the inheritance from Heron was so complete that it even concerned the end to which the machine was put. Heronian technology hung on for another century in various hands, until it became convenient to start building steam engines - which is to say, when the rapidly growing energy needs of nascent industrialization no longer could be met by watermills alone."
[8] There's even evidence of there being dissections of "condemned men" while they were still alive!
[9] It would not be seen again until Carl von Linné (Carolus Linnaeus).
[10] "We have seen, then, that the bases of modern evolutionism, namely the notions of mutation and natural selection, were both present in Hellenistic thought."
[11] "This is consistent with Heron's remark that an early automatic playlet merely showed, by way of motion, a face with blinking eyes - something that is of course easy to accomplish with an alternation of just two images. Heron also says that with still automata one can either show a character in motion, or a character appearing or disappearing."
[12] "The Hellenistic origin of the novel has long been obscured. It was thought that Greek-language novels first appeared in the late imperial age; this changed in 1945 when a papyrus was found in Oxyrynchus that dates from the first century B.C. and contains fragments of the
Novel of Nivus. Now many scholars think that the novel originated in the second century B.C."
[13] "In the early sixteenth century Thomas More wrote admiringly that in Utopia "vast numbers of eggs are laid in a gentle and equal heat, in order to be hatched", but incubators would remain a mere literary memory still long after that."
