The answer: it depends on what time period you're talking about. During the reign of Basil II (976-1025) provinces were divided into military-administrative districts known as themes. The themes were overseen by generals, who commanded an army specific to that theme. There were a few large landholders, the "nobility" if you will, but after a number of rebellions Basil II much reduced the power of the nobility and so a larger number of soldier-farmers emerged. Tax collectors came from the capital and usually collected from communities of farms. This differs from serfdom or other forms of peasant-lord relationship in that Eastern Roman peasants were nominally independent and not bound to the land. Of course, if your land was bought by a large landowner that changes and you are effectively a serf.

The Eastern Roman Empire was much more centralized than other kingdoms of the time, a holdover from the days of the old Empire. There was no formal nobility but there were powerful families that held high titles in the army and bureaucracy, some examples being the Skleroi, Doukoi, and Komnenoi clans.

Manzikert is often seen as a major watershed for the empire, but even before that large landholders began to regain the power they had lost during the reign of Basil II. By the ascension of Romanus IV Diogenes (1068-71) the thematic system was decayed. The soldier-farmer backbone of Basil II's army was broken, and so Romanus was forced to rely on mercenaries to bolster his ranks. After the defeat at Manzikert and subsequent loss of Anatolia to the Seljuqs, the thematic system totally collapsed and the Empire gradually decentralized. By the reign of Andronikos II Palaiologos (1282-1328) the Empire had effectively relapsed into a decentralized state of large landholders, serfs, and some small farming communities.

Source: A Short History of Byzantium by John Julius Norwich

Well first we would have to define "feudalism" and set a time period for comparison.

While it certainly wasn't a very uniform form of government, I would define "feudalism" as a system where large landholders, "lords", have jurisdiction over their holdings and the peasants on them. These lords then generally swear fealty to higher lords, who protect them in exchange for tax and/or military levies, which is how most feudal "states" are structured. This is known as a "bottom up" structure of government, since power comes from the smaller lords, who invest it in the larger lords, who invest it in the king.

For which time period we'll be comparing, my expertise lies mostly in the middle byzantine empire, so 1000 ad works well.

In western Europe, lets say France specifically, at this time, feudalism worked much as described in the above paragraph. Serfs, peasants legally bound to a fief, would work the land and fight in the armies when needed, and the holders of the land they lived on would take the income they didn't need and command said armies. The system of government was organized around these land holders, who swore fealty to larger lords (or were granted a holding in exchange for their loyalty and service), who then would swear fealty to the king. There were free men not part of the landed elite, mostly concentrated in cities, but these were a small minority compared to the serfs at this time. It is worth saying that being a serf, while not pleasant, wasn't nearly as bad as being a slave, and most lived relatively content lives.

The Byzantine empire at this time used a bureaucratic system, not that unlike modern ones, whose power rested on the shoulders of free citizen farmers. The central government was roughly divided into ministries, which managed a certain aspect of government, and whose heads were appointed by the Emperor. These ministries were generally led a Logothete, the highest rank of minister essentially, who managed them and appointed sub-ministers. The ranks of the Logothetes included:

The Logothete of the Course, who managed intelligence gathering and the public post, and also often functioned as prime minister in the macedonian period.

The General Logothete, manager of the general financial ministry and in charge of finance and revenue.

The Logothete of the Military, in charge of soldiers' pay and provisions

The Logothete of the Herds, who managed imperial estates

The Logothete of Special affairs, who managed the imperial treasury, factories, storehouses, and monopolies

The Logothete of the Praetorium, an aid to the eparch of Constantinople in charge of Judicial matters

The Logothete of the Waters, In charge of maintaining and building the public Aqueducts

The Logothete of the Household, which was something of an auxiliary official whose duties were normally Judicial in nature

In addition to the Logothetes, there were also other important upper bureaucrats, such as the Eparch of Constantinople, who managed the city, and the Sakellarios, who sometimes acted as a chief financial minister in place of the Logothete.

In addition to the bureaucratic hierarchy, there was also the Provincial hierarchy. The provinces of the Byzantine empire during this period were divided into themes, which also had a military function. These themes were headed by Generals, Strategoi, who were appointed by the Emperor, and had military and Judicial authority (though they were also appointed a judge to help assist them in this) inside of their theme. The themes were further sub-divided into sections called Tourma, which were co-administered by a Tourmarch, who was also a military officer. The Tourmarchs took on a more legal and administrative duty than the Strategoi, and were more local in focus. In this period, the Themes themselves were also divided into the "Roman" themes, the traditional ones, and the kind I was just describing, and the "Armenian" themes, which were junior to the Roman ones and much smaller. These Armenian themes, set up on the eastern border, contained just a fortress and the surrounding area, and were well suited to defense.

The Byzantine empire, in contrast to western Europe, was similar to modern states in that it had a "top down" system of government. The ministry of finance would draw taxes from the citizenry, which it would use to fund the administration and protection of the empire via salaries and budgets.

On the non-governmental level, the Byzantine empire was populated almost entirely by free citizen farmers, who payed taxes to the government and drove the Byzantine economy. In addition to these though, there was also the significant Urban population, which made up around 15% of the population if you only count cities with 10,000 people or more, as well as the aristocratic elite. Urban citizens, most of which were concentrated in Constantinople (which had a population of up to 600,000, out of around 13,000,000 , in this period) and the surrounding area, were largely craftsmen, fishers, merchants, civil servants, or employees of those mentioned, and made up a much larger percentage of the Byzantine population than they did in western Europe.

There were also the Aristocrats of the Byzantine empire. These were the upper classes of society, who got rich from their large estates or important offices. Among the aristocrats were what are known as the Provincial magnates, large and wealthy families of Anatolia with a military and civil pedigree. From the ranks of these families sprung many officers, bureaucrats, and even emperors. While these families were wealthy and influential, they didn't serve the same function as feudal lords, nor did they have the same rights. The magnates didn't have any more legal rights that the other citizens of the empire, and it was illegal for them to make tenants work without pay. Their lack of inherent authority was illustrated well when Basil II, the emperor in 1000 ad, broke their power by enacting legislation requiring that they pay the taxes of their poorer neighbors.

In conclusion, the Byzantine empire is administratively different from the feudal kingdoms of western Europe in almost every way. It's economy rested on free farmers instead of serfs, it had a strong central government compared to the easily breakable kingdoms of western Europe, and it employed a massive meritocratic bureaucracy instead of the hereditary lord focused government of the west.

Sources:

"A History of Byzantine State and Society" by Warren Treadgold

"The Byzantine Republic" by Anthony Kaldellis

"The Economic History of Byzantium" by numerous authors, edited by Angeliki E. Laiou