Why were the Ottomans able to conquer the (eastern and southeastern) parts of the Middle East where the Byzantines failed to expand?

If I understand your question correctly, you're comparing the Ottoman Empire with the Byzantine Empire of Late Antiquity, roughly a thousand years earlier, prior to its dramatic territorial losses to the Arab conquests of the seventh century - and not the post-seventh-century empire. Is that right? I think it's a worthwhile question, and wish more comparative studies of the Ottomans and Byzantines would be carried out. As it stands, they are fields which don't overlap much. Despite the empires' similar geographic span and significant continuities, comparison is not given much attention. I hope someone familiar with the Byzantine Empire of this era can contribute their own perspective.

Not being an expert on the Byzantines, I'll have to limit my answer to the issue of what allowed the Ottomans to maintain their rule in the regions under discussion, Iraq and the Arabian Peninsula. I'll also limit my answer chronologically to the early modern period, the sixteenth to eighteenth centuries. The first thing we should note is that these regions were very much on the periphery of the Ottoman Empire. In the Arabian Peninsula, the Ottomans had no control whatsoever outside of the coastal regions in the east (al-Hasa) and west (Hijaz & Yemen). In Iraq, the Ottoman presence was limited to major cities such as Mosul, Baghdad, and Basra, with most of the countryside (especially in the south) under the control of tribes often only nominally loyal to the Ottomans. This meant that Ottoman authority was extremely fragile and could be rapidly overturned in the case of military defeat or political instability. Safavid Iran was able to temporarily evict the Ottomans from Iraq by conquering Baghdad in 1622, while Yemen and al-Hasa were lost to revolt in 1636 and 1670 respectively, not to be regained for hundreds of years. Large parts of these regions were not under direct Ottoman rule at all: most of the Hijaz was under the authority of the Sharif of Mecca, while Basra was ruled by the autonomous Afrasiyab dynasty, which occasionally clashed with Ottoman governors of Baghdad (and also resisted the aforementioned Safavid invasion) from 1596 to 1668.

In the Hijaz, what allowed the Ottoman to maintain their authority and the loyalty of the Sharif of Mecca, aside from their raw military power, was their role in supplying the cities of Mecca and Medina with foodstuffs. The Hijaz, not a particularly bountiful region, had relied for centuries on imported food from Egypt for its sustenance. When the Ottomans conquered Egypt in 1517, they immediately took over the role of managing this crucial supply route from the Mamluk Sultanate. They also set about constructing a fleet in the Red Sea in order to keep the sea lanes safe for traders, supply convoys, and pilgrims, as the Portuguese had already begun to menace the region, unsuccessfully attacking the port town of Jidda in the same year that the Ottomans conquered Cairo. Finally, throughout this period the Ottomans invested significant resources into maintaining the security of the overland pilgrimage routes to the holy cities, which served to bolster the dynasty's legitimacy. Thus the inhabitants of the Hijaz benefited from Ottoman rule and were reliant on continued ties with them.

In Iraq, Ottoman rule was much more reliant upon the empire's military strength. The Ottomans were quicker than the Safavids in adapting gunpowder weapons to effective military use and also had a numerical advantage in most engagements. Logistically, control over the Tigris and Euphrates rivers also made it easier for Ottoman armies to operate in Iraq, although doing so put an immense strain on their resources due to the sheer distance from the core regions of the empire. Despite this, overall the Ottomans had a much greater capacity to mobilize their military resources than the Safavids, so the latter typically found themselves on the defensive in the Zagros mountains and southern Caucasus, where the terrain evened the odds, while Iraq with its flat plains and hostile tribes was left to the Ottomans. The Safavid occupation of Baghdad from 1623-38 was something of an abberation coinciding with Ottoman political instability and Safavid cohesion under Shah Abbas I. Nevertheless, neither the Ottomans nor the Safavids ever established anything like tight control over the region, which was always limited to the major cities. Iran under Nader Shah would attempt to conquer Iraq again in the 1730s, unsuccessfully besieging Mosul and Baghdad. Again it was Ottoman military power, along with the effective leadership of local governors who were then consolidating their influence, which thwarted the attempt. As for why the Ottomans would want to control Iraq in the first place, I would identify three primary considerations. The first was defensive, a desire to simply deny their Iranian rivals access to the region. Second, access to Persian Gulf trade which, while less significant than the Red Sea, was still a major commercial highway. Third, the prestige and legitimacy associated with control over the former capital of the Abbasid Caliphate, the tomb of Abu Hanifa (the namesake of the Hanafi school of law which the Ottomans followed), and the Shi'i shrine cities of Najaf and Karbala. There was a clear feeling that Baghdad should not be left in Shi'i hands, and the Ottoman war effort in the 1620s and '30s was spurred on by reports of Safavid massacres of the Sunnis of Baghdad - whether true or not, I don't know.

As the eighteenth century wore on, Iraq was more tightly integrated with the rest of the empire by the expansion of the system of tax-farming, which made revenue collection rights a purchasable commodity. While often characterized as "decentralization," this process also tied a wider range of provincial elites to the Ottoman center, as local powerholders (called ayan) could purchase tax-farming rights and thus link themselves to the Ottoman state, giving themselves a stake in continued Ottoman rule and political stability. For this reason this process is often referred to as the "Ottomanization" of provincial society. While governing power was thereby decentralized, the Ottoman presence in the provinces widened somewhat beyond the narrow military-political elite and made the empire's hold over these territories more substantial.