What led to suicide being seen as honourable in eastern countries (particularly Japan), and why is it dishonourable in western cultures?

TL;DR: Christianity

In Antiquity, suicide wasn't seen as dishonourable, and to many it was the preferred solution to death by other means. Socrates is probably the most notable person from this era and location to kill himself. It should be noted that Socrates wasn't a fan of suicide, but was okay with it in specific situations. Plato wrote down Socrates defense of his own suicide in The Apology.

http://classics.mit.edu/Plato/apology.html

It is very important to note that those who were against suicide during Antiquity generally were against it for philosophical reasons or for pragmatic reasons. Roman soldiers were looked down on for suicide for the same reason they were looked down on for deserting... their numbers were needed, and they were letting their country down by killing themselves. Of course, there were situations where a soldier committing suicide wouldn't have been dishonorable, if it was an alternate to surrender.

Brutus (et tu, Brute?) even killed himself to avoid capture, as detailed here by Plutarch:

Then, after earnestly entreating them to save themselves, he withdrew a little way in the company of two or three friends, among whom was Strato, who had been his intimate since they studied rhetoric together. 7 This man he placed nearest to himself, and then, grasping with both hands the hilt of his naked sword, he fell upon it and died. 8 Some, however, say that it was not Brutus himself, but Strato, who at his very urgent request, and with averted eyes, held the sword in front of him, upon which he fell with such force that it passed quite through his breast and brought him instant death.

http://penelope.uchicago.edu/Thayer/e/roman/texts/plutarch/lives/brutus*.html

In the Old Testament, there are a few references to suicide, including King Saul and one of the sons of King David.

4 Then Saul said to his armor-bearer, “Draw your sword, and thrust me through with it, lest these uncircumcised come and thrust me through, and mistreat me.” But his armor-bearer would not, for he feared greatly. Therefore Saul took his own sword and fell upon it. 5 And when his armor-bearer saw that Saul was dead, he also fell upon his sword and died with him.

1 Samuel 31:4-5

When we get to the New Testament, the only reference I could find to suicide was Judas hanging himself after betraying Jesus.

And he cast down the pieces of silver in the temple, and departed, and went and hanged himself

Matt 27:6

If you were a Christian, like most of the Western world, Judas was probably not the company you wanted to keep.

Once Christianity began taking over Europe, suicide became an excommunicable offense. Although there aren't any scriptures DIRECTLY relating to suicide as a sin, the Church was able to use some of the scriptures referring to living for God, and honoring God with our bodies, and how only God should take away life, as examples of suicide being a sin.

"Thou Shalt Not Kill" was used by St. Augustus in The City of God as the justification for suicide being a sin. If God created life, one was not to end life, even if said life was their own life, because that life belongs to God and only He can say when it ends.

Book 1: Chapter 17

And consequently, even if some of these virgins killed themselves to avoid such disgrace, who that has any human feeling would refuse to forgive them? And as for those who would not put an end to their lives, lest they might seem to escape the crime of another by a sin of their own, he who lays this to their charge as a great wickedness is himself not guiltless of the fault of folly. For if it is not lawful to take the law into our own hands, and slay even a guilty person, whose death no public sentence has warranted, then certainly he who kills himself is a homicide, and so much the guiltier of his own death, as he was more innocent of that offense for which he doomed himself to die. Do we justly execrate the deed of Judas, and does truth itself pronounce that by hanging himself he rather aggravated than expiated the guilt of that most iniquitous betrayal, since, by despairing of God's mercy in his sorrow that wrought death, he left to himself no place for a healing penitence? How much more ought he to abstain from laying violent hands on himself who has done nothing worthy of such a punishment! For Judas, when he killed himself, killed a wicked man; but he passed from this life chargeable not only with the death of Christ, but with his own: for though he killed himself on account of his crime, his killing himself was another crime. Why, then, should a man who has done no ill do ill to himself, and by killing himself kill the innocent to escape another's guilty act, and perpetrate upon himself a sin of his own, that the sin of another may not be perpetrated on him?

Book 1: Chapter 20

It is not without significance, that in no passage of the holy canonical books there can be found either divine precept or permission to take away our own life, whether for the sake of entering on the enjoyment of immortality, or of shunning, or ridding ourselves of anything whatever. Nay, the law, rightly interpreted, even prohibits suicide, where it says, You shall not kill. This is proved especially by the omission of the words your neighbor, which are inserted when false witness is forbidden: You shall not bear false witness against your neighbor. Nor yet should any one on this account suppose he has not broken this commandment if he has borne false witness only against himself. For the love of our neighbor is regulated by the love of ourselves, as it is written, You shall love your neighbor as yourself. If, then, he who makes false statements about himself is not less guilty of bearing false witness than if he had made them to the injury of his neighbor; although in the commandment prohibiting false witness only his neighbor is mentioned, and persons taking no pains to understand it might suppose that a man was allowed to be a false witness to his own hurt; how much greater reason have we to understand that a man may not kill himself, since in the commandment, You shall not kill, there is no limitation added nor any exception made in favor of any one, and least of all in favor of him on whom the command is laid! And so some attempt to extend this command even to beasts and cattle, as if it forbade us to take life from any creature. But if so, why not extend it also to the plants, and all that is rooted in and nourished by the earth? For though this class of creatures have no sensation, yet they also are said to live, and consequently they can die; and therefore, if violence be done them, can be killed. So, too, the apostle, when speaking of the seeds of such things as these, says, That which you sow is not quickened except it die; and in the Psalm it is said, He killed their vines with hail. Must we therefore reckon it a breaking of this commandment, You shall not kill, to pull a flower? Are we thus insanely to countenance the foolish error of the Manichæans? Putting aside, then, these ravings, if, when we say, You shall not kill, we do not understand this of the plants, since they have no sensation, nor of the irrational animals that fly, swim, walk, or creep, since they are dissociated from us by their want of reason, and are therefore by the just appointment of the Creator subjected to us to kill or keep alive for our own uses; if so, then it remains that we understand that commandment simply of man. The commandment is, You shall not kill man; therefore neither another nor yourself, for he who kills himself still kills nothing else than man.

http://www.newadvent.org/fathers/120101.htm

Augustus' influence on Christianity, and thus on the entire Western World, cannot be ignored. He essentially writes the book on why suicide is wrong, from a religious perspective.

Remember, unlike Socrates' and Plato's ethical reasons for being against suicide, the Christian reasons extend far beyond this life and into the next. If you committed suicide in ancient Greece or ancient Rome, you could be seen as dishonorable based on the context of your decision, but that decision only extended to your current life. If you kill yourself in the Middle Ages, or even now, that act will lead to you being punished for eternity, as you've sinned against God, and as it is the final act of your life, there's no asking for forgiveness.

This logic is presented by St. Thomas Aquinas:

because life is God's gift to man, and is subject to His power, Who kills and makes to live. Hence whoever takes his own life, sins against God, even as he who kills another's slave, sins against that slave's master, and as he who usurps to himself judgment of a matter not entrusted to him. For it belongs to God alone to pronounce sentence of death and life, according to Dt. 32:39, "I will kill and I will make to live."

St. Thomas Aquinas, Summa Theologiae, SECOND PART OF THE SECOND PART : L.63, C.6.

http://www.documentacatholicaomnia.eu/03d/1225-1274,_Thomas_Aquinas,Summa_Theologiae%5B1%5D,_EN.pdf

(warning, very long PDF)

In conclusion, the reason suicide is looked down upon so much in Western Cultures is because Western Society is intertwined with Christianity, and some of the most influential Christian philosophers were extremely firm on their stance that suicide is violating God's will, and is in direct violation of the Ten Commandments.

In Asia, and particularly Japan, Christianity never really got a foothold until modern times, and even now Christianity is extremely uncommon in Japan. The Japanese have a long and proud history and culture, and as most of them follow either Zen Buddhism, Shinto, or both, there isn't much room for Chistianity.