Interesting question.
First thing first, the Umayyad without the Abbasid revolution are in a very dangerous predicament. I typically give Islams immediate survival to the Abbasid period of calming of expansion and focusing it mainly around looting. The Umayyads had by this point began to exhaust itself, especially in the Caucus, Anatolia and Ifriqiya/Andalus. So I will give a scenario of what could happen had the Umayyads remained in the throne of Islam instead of the Abbasid throne.
1. The Byzantine powers by the 730s have already begun to have a resurgence in power. This mai my derives from the beginning weakness from the previously dominant Umayyad forces as a consequence of their many territorial wars which pulled and gave strain to the empire. This was further increased by the fact that the Umayyad was based out of Dimashq which made it more dépendant on winning wars in Europe, which was becoming more and more difficult.
By the 740s, I would argue that the Umayyad will never take Constantinople and without the change of rule, and continued Umayyad negligence, the Caliphate will be overtaken in relative power by Byzantium and slowly if they do not act, Byzantium will begin to turn the naval war into a one sided Byzantine affair and push back the Arabs from the fringes of Anatolia.
Notice too, that the Umayyads do not benefit from the boost of Baghdad and the inclusion of ethnic groups not Arab as the Abbasid did and still posses an even more rebellious Iran. Without the Abbasid changes to the fundamentals of being a Muslim, Iran likely remains Zoroastrian and the possibility of a return to Zoroastrian rule is not out of the question. Especially this could be the case if the powerful Alid groups in Iraq side with a Zoroastrian strong man, perhaps an alternate Ya'qub ibn Layth al-Saffarid or alternate al-Afshin.
If such a Zoroastrian revival does occur, that could leave the Umayyads in control over Arabia, Levant and North Africa.
Al-Andalus is hard to read. It could be the case that in this scenario, it becomes the Pakistan of the tl. With Iran out of the way, thus removing India as a place to raid; France and Europe become the next favorite for Caliphal sanctioned raiding. However, I do not see the Umayyads without Iraq and Iranveing able to win in a long war with Byzantium in the long run, the Abbasids only just draws the Byzantines and it controlled all of the Islamic East and much of Ifriqiya (east of Algiers).
Another possible effect, is less Islamic conversion in the Levant. With the extremely pro Arab identity of the Umayyad, it could be the case that the majority of thé Levant remains Christian for much longer and by extension, Syriac benefits greatly from this.
Also, the possibility to another Khawarij revolt is all too likely. If another major revolt like the Berber revolt occurs again; I am afraid the Umayyad would not survive save an incredible General.
Sorry for such a weak response, I have only just awoken.