Person 'A' is a child/rapist/killer. Person 'B' is very kind and never intentionally hurt anyone. They both die without accepting Jesus as Saviour. Will there be any difference with regards to the degree of punishment they will receive after death?

Luke, chapter 12, tells the parable of one who has left the care of the estate to his servants until he returns. He returns unexpectedly and finds some faithfully serving and others not serving at all. He rewards the faithful servants. Then we read:

Luke 12:46-48 [46] The lord of that servant will come in a day when he looketh not for him, and at an hour when he is not aware, and will cut him in sunder, and will appoint him his portion with the unbelievers. [47] And that servant, which knew his lord's will, and prepared not himself, neither did according to his will, shall be beaten with many stripes. [48] But he that knew not, and did commit things worthy of stripes, shall be beaten with few stripes. For unto whomsoever much is given, of him shall be much required: and to whom men have committed much, of him they will ask the more.
See All... The lord of that servant will come in a day when he looketh not for him, and at an hour when he is not aware, and will cut him in sunder, and will appoint him his portion with the unbelievers. And that servant, which knew his lord's will, and prepared not himself, neither did according to his will, shall be beaten with many stripes. But he that knew not, and did commit things worthy of stripes, shall be beaten with few stripes. For unto whomsoever much is given, of him shall be much required: and to whom men have committed much, of him they will ask the more.

The servant who knew the Lord's will and did it not will receive many stripes, while the servant who sins but in ignorance will receive few stripes. This is definitely a variation in judgment. Do not let the idea that they are called servants fool you. They are appointed their portion with the unbelievers. They are not true servants. In another instance, the Bible compares the judgment received by people of different places and times:

Matthew 11:23-24 [23] And thou, Capernaum, which art exalted unto heaven, shalt be brought down to hell: for if the mighty works, which have been done in thee, had been done in Sodom, it would have remained until this day. [24] But I say unto you, That it shall be more tolerable for the land of Sodom in the day of judgment, than for thee.
See All... And thou, Capernaum, which art exalted unto heaven, shalt be brought down to hell: for if the mighty works, which have been done in thee, had been done in Sodom, it would have remained until this day. But I say unto you, That it shall be more tolerable for the land of Sodom in the day of judgment, than for thee.

Though Capernaum will be brought down to hell with Sodom, it till be more tolerable in the day of judgment for Sodom than for Capernaum. Evidently, the light provided by Jesus Christ gave much greater responsibility to the New Testament city. By rejecting greater light, they received greater condemnation.

James 3:1My brethren, be not many masters, knowing that we shall receive the greater condemnation.
See All... says that the masters (the teachers of the word) "shall receive the greater condemnation. In addition, Jesus gave this warning to the scribes and Pharisees:

Matthew 23:13-14 [13] But woe unto you, scribes and Pharisees, hypocrites! for ye shut up the kingdom of heaven against men: for ye neither go in yourselves, neither suffer ye them that are entering to go in. [14] Woe unto you, scribes and Pharisees, hypocrites! for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation.
See All... But woe unto you, scribes and Pharisees, hypocrites! for ye shut up the kingdom of heaven against men: for ye neither go in yourselves, neither suffer ye them that are entering to go in. Woe unto you, scribes and Pharisees, hypocrites! for ye devour widows' houses, and for a pretence make long prayer: therefore ye shall receive the greater damnation.

These scribes and Pharisees do not go into heaven and they hinder others so that they do not enter heaven either. However, the position and place of knowledge given to the Pharisees create in them a greater responsibility than those they deceive. Therefore, they receive "the greater damnation." This could only point to a greater punishment in hell.

Although there is no verse that directly deals with the various degrees of punishment in hell, there is much incidental evidence like that given above. I think we can rest assured that there will be various levels of judgment in hell. However, that should be no comfort to the modern rejecter of Christ who lives a fairly decent lifestyle. Much of the greater punishment is reserved for that one who knows all the dangers of hell and the blessings of heaven and who still turns against God in unbelief. May those who have rejected the Lord Jesus Christ trust in Him as Saviour today.