What if Islam never came along? What would the world be like?
Well, first of all; it would actually be pretty hard to predict what the world would be like without Islam, because any scenario without Islam has a lot of variables that are hard or even impossible to predict.
However, a few things are quite likely to happen in this scenario;
First of all, there would be no Dark Ages in Western Europe, but there would not be a Renaissance either.
The two main factors that led to the Dark Ages in Western Europe are the collapse of the Western Roman Empire and the resulting political instability in Western Europe (the barbarian invasions, the destruction of the established and more or less Romanized barbarian kingdoms by Justinian I), the initial Muslim invasion (which conquered Roman Africa and Spain) and Saracen piracy (which not only includes ordinary piracy, but also a good number of what can only be described as militairy operations with the purpose of plunder instead of conquest).
Prior to the Muslim invasion, Western European civilisation was still pretty much centered around the (Western) Mediterranean, and it's sphere of influence included parts of North Africa. Consequently, there was relatively much cultural interaction and exchange between the Latin West and the Greek East.
And when the Muslims invaded in OTL, they basically occupied almost every civilized part of Western Europe save for mainland Italy (the short-lived Saracen colonies in Calabria and Apulia aside). What was left of Christian Western Europe were the less developed nations in the north, and Italy, which was in decline at this point.
So without the Muslim conquest and Saracen piracy, Christian Western Europe would have retained its main centres of population, culture and civilisation, would have remained centered around the Western Mediterranean, its economy would not have been hurt by Saracen piracy, and it would have more contact with other civilisations.
Western Europe would still be in a decline for a while, but it would recover as soon as stable nations would have been established in the area.
And its also very likely that Christianity, along with European influences, will spread in the Berber kingdoms of Northwest Africa. And that could very well lead to direct or indirect contact between Western Europe and the civilisations and kingdoms of Sub-Saharan West Africa.
However, without Islam, the factors that set off the Renaissance in OTL would not have existed (no Islamic sciences and philosophy to boost Western European sciences, no Dark Ages in Western Europe), so the development of civilisation in Medieval Europe would be far more gradual than it was in OTL.
And wether scientific developments and the development of civilisation in general would be faster, slower or roughly similar to OTL is anybody's guess. IMHO there are far too many unpredictable variables in this scenario to make any real predictions about that.
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As for the Middle East and the Byzantine Empire: provided that the Sassanids do not come back strong, there will be a bit of a political vacuum in the Middle East. Because of this, it is likely that there will be some effectively independant kingdoms and principalities between the Byzantine and Persian empires.
There will almost certainly be at least one Armenian kingdom, and if the Sassanid Empire collapses, it is quite possible that other states will emerge as well, perhaps domains carved out by Persian generals or princes, or Arab or Aramaic warlords.
We could see a "warring states"-period in the Middle East, which would be quite interesting.
And these developments will affect Byzantium as well; many Armenian aristocrats fled to the Byzantine Empire after the Caliphate subjugated Armenia in OTL, and there were in fact several Byzantine emperors of Armenian origin. Those emperors would certainly be butterflied away in this scenario, and that will deeply affect the Byzantine Empire.
What would happen in the non-Orthodox (non-Chalcedonian) parts of the Byzantine Empire after the Byzantine-Persian war is rather hard to say. There are a lot of possebilities for succesful rebellions et al during this period, and it would take years, if not decades, for the Byzantines to properly reassert their authority over the recently regained territories.
Growing Nubian influence in Egypt is a possebility, allthough this might be a bit trickier and more complicated than I used think; the Nubian kingdoms Alwah and Nobotia followed the miaphysite Coptic Church, but a few months ago, I found out that the Church of the Nubian kingdom of Makuria followed the Chalcedonian dogma, and that the miaphysite factions only dominated there after the Byzantine Emperor stopped appointing Melkite patriarchs of Alexandria due to the Muslim conquest of Egypt.
And as long as the Chalcedonian Church of Makuria remains in direct contact with the other Chalcedonian Churches through the Melkite Patriarch of Alexandria, it is rather unlikely that the Makurians would ever switch to the miaphysite Coptic Church.
Nonetheless, if a succesful Coptic rebellion would make Egypt independant and bring it under Coptic rule, then it is safe to say that there will be no more Melkite patriarchs of Alexandria, so it is still possible for Makuria to become miaphysite in this scenario.
But the Byzantines would propably hold on to Syria and Palestine, as those areas were easier to reach from the Byzantine core territories and had a smaller population than Egypt.
And without a Muslim invasion to be worried about, the Byzantines would propably be able to handle the second great Donatist revolt that was raging in Byzantine Africa during the 7th century, so they would propably hold on to Africa as well.
The Turkish migrations into the Middle East would most propably still happen, and depending on the developments within the Byzantine Empire, a Turkish invasion of Anatolia is still possible (maybe not very likely, but still possible).
However, it is very likely that these Anatolian Turks would be converted to Orthodox Christianity and (partially) assimilated relatively soon (pretty much like the Slavs on the Balkan), and the best case scenario for such Orthodox Anatolian Turks would be that they are unified into a single, lasting empire by one of those Turkish tribes. I.e. a Bulgaria-analogue in Anatolia.
But things could get interesting if those Turkish invaders (or at least their ruling clans) would have converted to the Church of the East (aka. Nestorianism), one of the Miaphysite Churches (Armenian or Syriac Orthodox), or perhaps even to a surviving or new Gnostic religion (there were a few strong Gnostic sects in early Medieval Armenia, and Manicheism was still strong during the early Medieval ages).
However, even if, say, a Nestorian Turkish horde would conquer Anatolia, then there's a good chance that this Turkish empire will become a lot like early Visigothic Spain or Ostrogothic Italy; a relatively small elite of Nestorian Turks who rule over a large Greek and Orthodox population, with a fairly high propability of the elite becoming Orthodox.
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And as for the most controversial issue; the development of sciences without Islam:
Advanced sciences would definitely exist in the Middle East, even without Islam, as there was already a long tradition of learning and even translating ancient works of science and philosophy in other languages. The universities of the Abbasid Caliphate were modeled on the universities of the Sassanid Empire, and the invaluable contributions of non-Muslim (and in many cases Christian) scientists and translators to 'Islamic' sciences are often overlooked.
However, for those sciences to develop on a similar scale as they did in the OTL Abbasid Caliphate and contemporary Muslim states, you'll need a fairly large and powerful empire that actively supports the development of these sciences.
The most likely candidate for such an empire would be a resurgent Persian (but not neccesarily Sassanid) Empire, allthough there are also more exotic options, like a non-Islamic Arab empire (something akin to the Ghassanids or Lakhmids, but then on a bigger scale) or even an Aramaic empire.
But it is nonetheless quite likely that there will be more significant "scientific traditions" in a world without Islam, even though most of those scientific traditions would not be as great as the OTL Islamic scientific tradition.
It seems rather likely that the Western European (i.e. Latin) scientific tradition would interact more with the Greek (i.e. Byzantine) scientific tradition and quite possibly the Aramaic scientific tradition(s) (pilgrimage to Jerusalem could very well have brought Christians from Western Europe in contact with the scientific traditions of the Middle East, and I recall that a Syrian became the Pope of Rome in the 8th century in OTL).
However, it would most propably take a lot longer for eastern (Middle Eastern and beyond) scientific influences and discoveries to penetrate into Western Europe, as these eastern influences cannot be introduced directly into Europe through Islamic sciences here.
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It's just an interesting sidenote, but it is also possible that a distinct Ethiopian scientific tradition would form in this scenario. In OTL, Ethiopia lost its coastal territories to the Muslims fairly soon after the rise of Islam, and that was followed by an Ethiopian dark age of which fairly little is known.
But in this scenario, Ethiopia, or more precisely, Axum, would remain a naval power, and it would remain in direct contact with Egypt, the rest of the Eastern Mediterranean, the various nations of the southern Arabian Peninsula, East Africa, and India.
And it would be quite interesting to see an Ethiopian scientific tradition that combines strong Greek, Coptic, and Indian influences... (it's perhaps a bit of a stretch, but it is possible that the concept of zero is introduced in Europe from India through Ethiopia in this scenario, instead of from India through the Muslim states, like in OTL)