What problems made Rome weaker?
This is a loaded question. Scholars will have different opinions. More importantly, there are probably a confluence of factors that weakened Rome.
First, we can mention external problems and foes. As Rome continued to expand, there were too many borders to defend. This meant that Rome had more foes. For example, the Gauls were never fully defeated by Caesar. And in time, the Gauls and the Germanic tribes across the Danube would constantly be a threat. This is not even to mention the Roman foes in the East, such as the Parthians.
Second, there were also internal problems. The Romans were always competing against each other. Civil Wars, undoubtedly, created havoc and weakness among the Romans.
Third, the Roman Empire became too complex. When societies become too complex, they may begin to implode by the sheer weight of their own success.
Fourth, there is something to be said about luxury. Rome might have lost some of its hard edge by centuries of luxurious living. Being entitled usually weakens a group. The fact that the Romans used more and more mercenaries points in this direction.
Finally, Rome also "separated." We must keep in mind that Constantinople and the East began to gain in importance. Constantinople stood until 1453. Hence, resources were diverted to the East.