I interpret the question to mean how is the decline of the holy Roman (western) empire in 476 CE different from the decline/tribulations of its offshoot the Byzantine (eastern Roman) in the 9th century.
The Byzantine empire practiced Christianity only whilst the early Roman (western) empire, also called the Holy Roman empire, observed all religions.
The Western Roman empire suffered a gradual decline caused by a number of factors: (1) christianity took precedence over empire growth and domination (2) loss of morality (3) economic decline and chronic unemployment (4) implosion. Diocletian declared that Rome should stop expanding and the empire lost its focus. These four factors were thought to result from Rome lacking a powerful executive head or leadership and relying on a senate system which led to internal conflict with its own military.
The decline of the Byzantine (eastern Roman empire) - which considered itself a continuation of the Holy Roman empire was mainly military as it came under attack from Arab muslims.