The eastern and Western halves had been different from the very beginning. In the Eastern Mediterranean, in which I include Roman Greece, Asia Minor, Syria, and Egypt, the Roman encountered and conquered the Hellenistic world--that is, the series of successor states to Alexander's conquest. These states had a Greek elite, and urbanized culture, and clearly organized political institutions. This made the conquests have something of a dual aspect: the Hellenistic states fielded large, organized armies led by a clear political elite, which seems like it would make things more difficult, but in reality this was exactly the form of warfare the Romans were good at. And, once they inevitably smashed the Hellenistic armies, converting pre-existing administration to their needs was fairly easy. The conquest of the East was therefore characterized by large but discrete battles followed by a smooth administrative transfer and a fairly easy rule.

The West was different. Rather than facing a series of clearly organized urban states, the Romans faced a set of smaller scale polities characterized by shifting relations and fluid relative hierarchies. Caesar's conquest of Gaul is a good example: Caesar defeated his enemies on the field and subdued their leaders, but a Vercingetorix could arise and make previous arrangements irrelevant. Or more starkly, in Britain Caesar could consistently defeat his foes on the battlefield, but as soon as he tried to solidify his conquests through tribute arrangements this met British reality and dissolved. This is why the conquest of Spain took far longer than the conquest of the entire Eastern Mediterranean: the Romans could defeat the Spanish on the field (although with rather less consistency) but could not make the new political order permanent. Thus, the conquest of the West was longer, harder, bloodier and more destructive than the conquest of the East.

Naturally, there is a myriad of other differences to discuss, and exceptions to what I discussed (North Africa is a glaring one) but I think it is vital to remember than the situation at conquest was drastically different.

A few points to add to what Tiako already said. First, strategically, the eastern empire had two borders to protect, the Danube and the frontier with Persia, in contrast to the West, which had only the long Rhine-Alps border. The need to do both these things at once was the defining military objective of the eastern empire, and it would have a considerable impact on the history of the empire as a whole. On a related note, those who argue that Persia was a rival superpower, rather than a major power that was nonetheless firmly in Rome's shadow, would also see the Persian border and the extensive Roman investment in defending it as a very important distinction between the empire's halves.

Administratively, I suppose the main point would be that in the last decades of the Western Empire's history, the state had almost no authority in increasingly large areas of the empire's supposed territory, due to the barbarian nations that it couldn't dislodge (and partly because of banditry, too). The authorities in Constantinople, due to the East's sustained territorial integrity, had a lot more control over their realm. Other than that, though, the administrative systems were pretty much the same: in both halves, praetorian prefectures contained several dioceses which contained numerous provinces. And both halves involved a large imperial court which in the late empire produced the powers behind the throne who actually controlled the state.