Dying Vs Perishing
Dying
And it came to pass, that the beggar died, and was carried by the angels into Abraham's bosom: the rich man also died, and was buried; (Luke 16:22)
The Greek words for "died" in this passage are ἀποθανεῖν (transliterated apothanein) and ἀπέθανεν (transliterated apethanen).
Perishing
And Jesus answering said unto them, Suppose ye that these Galilaeans were sinners above all the Galilaeans, because they suffered such things? I tell you, Nay: but, except ye repent, ye shall all likewise perish. Or those eighteen, upon whom the tower in Siloam fell, and slew them, think ye that they were sinners above all men that dwelt in Jerusalem? I tell you, Nay: but, except ye repent, ye shall all likewise perish. (Luke 13:2-5)
The Greek word for both occurrences of the word "perish" in this passage is ἀπολεῖσθε (transliterated apoleisthe).
For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. (John 3:16)
The Greek word for "perish" in this passage is ἀπόληται (transliterated apoletai).
The Lord is not slack concerning his promise, as some men count slackness; but is longsuffering to us-ward, not willing that any should perish, but that all should come to repentance. (2 Peter 3:9)
Death as a Result of Sin
The Original Sin
And the LORD God commanded the man, saying, Of every tree of the garden thou mayest freely eat: But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die. (Genesis 2:16-17)
And when the woman saw that the tree was good for food, and that it was pleasant to the eyes, and a tree to be desired to make one wise, she took of the fruit thereof, and did eat, and gave also unto her husband with her; and he did eat. (Genesis 3:6)
And unto Adam he said, Because thou hast hearkened unto the voice of thy wife, and hast eaten of the tree, of which I commanded thee, saying, Thou shalt not eat of it: cursed is the ground for thy sake; in sorrow shalt thou eat of it all the days of thy life; Thorns also and thistles shall it bring forth to thee; and thou shalt eat the herb of the field; In the sweat of thy face shalt thou eat bread, till thou return unto the ground; for out of it wast thou taken: for dust thou art, and unto dust shalt thou return... And the LORD God said, Behold, the man is become as one of us, to know good and evil: and now, lest he put forth his hand, and take also of the tree of life, and eat, and live for ever: Therefore the LORD God sent him forth from the garden of Eden, to till the ground from whence he was taken. So he drove out the man; and he placed at the east of the garden of Eden Cherubims, and a flaming sword which turned every way, to keep the way of the tree of life. (Genesis 3:17-19, 22-24)
New Testament
Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned: (Romans 5:12)
For the wages of sin is death; but the gift of God is eternal life through Jesus Christ our Lord. (Romans 6:23)
Merriam Webster's Definitions
Die: to pass from physical life
Perish: to become destroyed or ruined
Note that Merriam Webster provides other definitions of these two words; however, these listed definitions are most pertinent to what these words mean in Scripture.
Grieving Is Scriptural
Joseph's Father
And Joseph fell upon his father's face, and wept upon him, and kissed him. And Joseph commanded his servants the physicians to embalm his father: and the physicians embalmed Israel. And forty days were fulfilled for him; for so are fulfilled the days of those which are embalmed: and the Egyptians mourned for him threescore and ten days. (Genesis 50:1-3)
Moses
So Moses the servant of the LORD died there in the land of Moab, according to the word of the LORD. And he buried him in a valley in the land of Moab, over against Bethpeor: but no man knoweth of his sepulchre unto this day. And Moses was an hundred and twenty years old when he died: his eye was not dim, nor his natural force abated. And the children of Israel wept for Moses in the plains of Moab thirty days: so the days of weeping and mourning for Moses were ended. (Deuteronomy 34:5-8)
King Saul
Then said Saul unto his armourbearer, Draw thy sword, and thrust me through therewith; lest these uncircumcised come and thrust me through, and abuse me. But his armourbearer would not; for he was sore afraid. Therefore Saul took a sword, and fell upon it. And when his armourbearer saw that Saul was dead, he fell likewise upon his sword, and died with him. So Saul died, and his three sons, and his armourbearer, and all his men, that same day together...Then David took hold on his clothes, and rent them; and likewise all the men that were with him: And they mourned, and wept, and fasted until even, for Saul, and for Jonathan his son, and for the people of the LORD, and for the house of Israel; because they were fallen by the sword. (1 Samuel 31:4-6, 2 Samuel 1:11-12))
Uriah the Hittite
And it came to pass, when Joab observed the city, that he assigned Uriah unto a place where he knew that valiant men were. And the men of the city went out, and fought with Joab: and there fell some of the people of the servants of David; and Uriah the Hittite died also...And when the wife of Uriah heard that Uriah her husband was dead, she mourned for her husband. (2 Samuel 11:16-17,26)
Absalom David's Son
And the king said unto Cushi, Is the young man Absalom safe? And Cushi answered, The enemies of my lord the king, and all that rise against thee to do thee hurt, be as that young man is. And the king was much moved, and went up to the chamber over the gate, and wept: and as he went, thus he said, O my son Absalom, my son, my son Absalom! would God I had died for thee, O Absalom, my son, my son! And it was told Joab, Behold, the king weepeth and mourneth for Absalom. And the victory that day was turned into mourning unto all the people: for the people heard say that day how the king was grieved for his son. And the people gat them by stealth that day into the city, as people being ashamed steal away when they flee in battle. But the king covered his face, and the king cried with a loud voice, O my son Absalom, O Absalom, my son, my son! (2 Samuel 18:32-19:4)
Sons and Daughters of Job
While he was yet speaking, there came also another, and said, Thy sons and thy daughters were eating and drinking wine in their eldest brother's house: And, behold, there came a great wind from the wilderness, and smote the four corners of the house, and it fell upon the young men, and they are dead; and I only am escaped alone to tell thee. Then Job arose, and rent his mantle, and shaved his head, and fell down upon the ground, and worshipped, And said, Naked came I out of my mother's womb, and naked shall I return thither: the LORD gave, and the LORD hath taken away; blessed be the name of the LORD. (Job 1:18-21)
Stephen
And they stoned Stephen, calling upon God, and saying, Lord Jesus, receive my spirit. And he kneeled down, and cried with a loud voice, Lord, lay not this sin to their charge. And when he had said this, he fell asleep...And devout men carried Stephen to his burial, and made great lamentation over him. (Acts 7:59-60, 8:2)
Even Jesus Grieved
These things said he: and after that he saith unto them, Our friend Lazarus sleepeth; but I go, that I may awake him out of sleep. Then said his disciples, Lord, if he sleep, he shall do well. Howbeit Jesus spake of his death: but they thought that he had spoken of taking of rest in sleep. Then said Jesus unto them plainly, Lazarus is dead...The Jews then which were with her in the house, and comforted her, when they saw Mary, that she rose up hastily and went out, followed her, saying, She goeth unto the grave to weep there. Then when Mary was come where Jesus was, and saw him, she fell down at his feet, saying unto him, Lord, if thou hadst been here, my brother had not died. When Jesus therefore saw her weeping, and the Jews also weeping which came with her, he groaned in the spirit, and was troubled, And said, Where have ye laid him? They said unto him, Lord, come and see. Jesus wept. Then said the Jews, Behold how he loved him! And some of them said, Could not this man, which opened the eyes of the blind, have caused that even this man should not have died? Jesus therefore again groaning in himself cometh to the grave. It was a cave, and a stone lay upon it. (John 11:11-14, 31-38)
Those Who Died in Christ
But I would not have you to be ignorant, brethren, concerning them which are asleep, that ye sorrow not, even as others which have no hope. For if we believe that Jesus died and rose again, even so them also which sleep in Jesus will God bring with him. For this we say unto you by the word of the Lord, that we which are alive and remain unto the coming of the Lord shall not prevent them which are asleep. For the Lord himself shall descend from heaven with a shout, with the voice of the archangel, and with the trump of God: and the dead in Christ shall rise first: Then we which are alive and remain shall be caught up together with them in the clouds, to meet the Lord in the air: and so shall we ever be with the Lord. Wherefore comfort one another with these words. (1 Thessalonians 4:13-18)