The mid-late Byzantine Empire is often characterized as a long but steady decline toward an inevitable collapse, sometimes slowed by competent emperors, but still a foregone conclusion well before the final fall of Constantinople in 1453 (or, if you count Trebizond as a true heir to the Byzantine legacy, then it’s fall in 1461).

Yet this view mischaracterizes the actual situation. Byzantium was in no way predestined to fall, and it experienced high and low points like any other empire. Similarly, the Ottomans had various periods of major success and dismal failure throughout their centuries-long history.

There were key differences between the two at the time they both existed, however, and one of the most important of those is demographics. After the Byzantine defeat by the Seljuk Turks at the Battle of Manzikert in 1071 AD, the Byzantines all but abandoned the Cappadocian plain in central Anatolia. Turkic tribes migrating from Central Asia (including the Seljuks, among others) took advantage of this vacuum and settled en masse. While ethnic Greeks continued (up to the 20th century) to inhabit toe coastal cities of Anatolia, the demographics of the central portion of the peninsula was shifting from Greek Christian to Muslim Turkic, a transition that would not reverse in the coming centuries. Turkic power on the peninsula grew at the expense of Greek power.

Still, though, Byzantine defeat was not inevitable. The Byzantines enjoyed major successes against the Turks in the coming centuries, especially with the aid of Western Crusaders. However, the Crusaders brought a mixed legacy for the Byzantines, initially restoring Christian power in the Levant, but also spearheading the disastrous Fourth Crusade, which many historians (and I) believe was the true nail in the Byzantine coffin.

Fast forward to the 14th century, when the Ottomans (just one of many Turkic tribes in Anatolia) were enjoying their greatest expansion, they were also aided by major demographic differences. What was once the center of the Byzantine Empire, and a bastion of Christian Greek culture, had become Turkish, putting even more pressure on a Byzantine empire already stretched thin. At this point, the demographic changes were firmly in place.

For further reading, I highly recommend Edward Luttwak’s Grand Strategy of the Byzantine Empire. It has a mostly military focus, but also accurately describes the changes occurring at this time. Another great one is John Julius Norwich’s Short History of the Byzantine Empire, an abridged 3-volume work that is meticulously researched and well-written. Both are great starting points for learning about all things Byzantine.

In addition to Cato The Elder's points, the Byzantine empire during the thematic period (650~ - 1015~) was also fundamentally defensive in nature.

In theme system was a military administrative system designed in the face of the arab conquests with the goal of making it as hard as possible for arab, and later alan and bulgar, armies to raid or invade Anatolia. During this period the empire was divided into military provinces, "themes", from which the system gets its name. Each theme was governed by a "strategos" (general) who oversaw recruitment and training of soldiers, maintenance of fortifications, and commanded the local army when necessary.

The main difference between the old provincial system and the new thematic one was in the nature of the recruitment, maintenance, and use of soldiers. Theme soldiers were locals who were rented state land at a very low price in return for a small wage and serving in the army when necessary. The theme system made these soldier both plentiful and relatively cheap for the state to maintain, but it tethered the essentially part time soldiers quite firmly to their local theme. This wasn't a major drawback in the face of the arab raids, but it heavily limited the percentage of the army that could be put to use in offensive campaigns, hence why the byzantine empire barely expanded during most of the thematic period.

In addition to the thematic troops, there were also the "Tagmata", professional troops under the direct command of the emperor stationed in and around Constantinople. In the early theme system these forces were very small compared to those of the themes, but they grew exponentially in size during and after the economic recovery of the 9th century (going from comprising around 1/12th of the army to 1/5th). These forces were further strengthened by the 10th century military emperors, and they were what made the massive campaigns of figures such as Nicephorus Phocas and Basil II possible. By the early 11th century the theme system was becoming obsolete as the border of the empire moved further and further away from the traditional frontier, and was mainly kept alive to augment the tagmatic troops which were now stationed all across the empire. The theme system was finally put out of commission during the civil wars and economic upheaval of the mid 11th century. The byzantine empire didn't really have much of a chance to use this new offensive army before they were defeated by the Turks and Anatolia was taken, obliterating their army and manpower pool. The komnenian restoration was a brief revival characterized by the byzantine empire going back on the offensive using these professional field armies, but the following horribly incompetent Angelos dynasty and the sack of Constantinople in 1204 ensured that most progress was lost.

I've gotten a bit off track here, but my main point was that the middle byzantine empire was very defensive from 650-950, and was struck by disasters that stopped it from realizing the full potential of its offensive military from 950-1204. The ottomans on the other hand had an offensive army from the beginning, as well as competent leaders to use it and vulnerable enemies to use it against. Had the byzantine empire been luckier in the 11th century or the komnenian restoration had continued past manuel, it might have been expanded in a similar way to the ottoman empire (though religion and geography would be a much bigger burden on the Byzantines than the ottomans).

For one last quick note I would say that the Byzantines didn't really "run out of steam". The byzantine empire was fighting tooth and nail against much larger neighbors for the majority of its existence, so the static borders which can be seen for 300 years after the Arab conquests can be seen more as a great achievement in and of itself as opposed to a lack of achievement.