True the Roman Empire was divided into the Eastern and Western Empires, and were "technically" the same Empire with an Emperor in Rome, or increasingly on the front or in Ravenna, and the Emperor in Constantinople. Laws passed in either Empire were binding in both Empires, as was currency. The Eastern Empire was situated better by both position and wealth/resources. The Eastern Empire was thus better able to withstand many of the internal and external pressures that the Western Empire could not. Also the Roman Army was not what it had been in previous times. It had been divided into a mobile, highly trained Field Force, and a less well trained Frontier Force guarding the borders. As the Field Forces were depleted by near constant civil wars, the generals and emperors were forced to move the Frontier Forces into combat, leaving the borders with less and less defenders.
Add in the decreasing tax base in the Western Empire, and you can see why things were continuing to decline. The government began to issue edicts tying people to the professions of their fathers, which caused a lack of motiviation for people to try to better themselves, so the tax base continued to dwindle just as more money was needed for national defense. The Army was forced to start impressing men and boys into the ranks, and many would try to avoid being drafted by cutting off their right thumb so they couldn't hold a sword. Also, many people ran to the estates of the rich to avoid taxes and military service, and in return for their complete servitude, the rich would protect them, thus depriving the government of tax revenue and manpower.
The pressures of the various tribes cannot be underestimated, but the Romans had many times defeated superior forces easily, but with the manpower and revenue shortages, and a general apathy among the populance, the government was powerless to resist their foreign invasions. Once the tribes were on Roman soil, the Romans failed to Romanize them as they had others, leaving distinct cultures intact within their borders. These potential sources of revenue and manpower that could have revived the Western Empire were instead goaded into becoming enemies by the actions of incompetent leaders of the government and military. The results were disasterous, with the Roman Army being virtually wiped out over a short period of years, leaving the government to rely on mercenaries to protect the Empire, mercenaries they could ill afford to pay.
Add to the mix the relationship between the rulers of the Eastern and Western Empires. At best of times there was some degree of cooperation, but those times were not the norm, and usually the two rulers and their Praetorian Prefects were plotting against each other. The true wonder is why the Western Empire lasted as long as it did.
While the Western Empire ceased to exist in AD 476 with the removal of Romulus Augustulus from the throne, the Eastern Empire continued until 1453 until the fall of Constantinople to the Ottoman Turks. Since Odovacar returned the Imperial Regalia to the Emperor Zeno in the Eastern Empire, the division was, official, over, although the west was of course lost. Given that Nepos, who had been nominated by the Eastern Emperor Leo (457-474), still reigned, albeit in exile, there was no question of Romulus ever receiving recognition from the east.