My self-chosed flair in /r/badhistory is "Thomas Becket needed killin'", so cum grano salis:
W. L. Warren's answer to "who ultimately won" might fairly be characterised as "pretty much everybody involved".
The church was still evolving and extending its canon law, and trying to extend its jurisdiction and power. But the fervent days of the Gregorian controveries were over: the church was generally recognizing the usefulness of coöperation and of having some regard for secular interests. After the justifiable homicide murder, it should be noted that the pope abhorred the deed, but did not excommunicate King Henry II (just prohibited him from entering a church) and merely confirmed the existing interdict on Henry's Continental domains only that had already been imposed by the archbishop of Sens. Instead, he seemed to be convinced in part by the king's own reported abhorrence and penance, and by the arguments of a partisan bishop of Thomas, and so sent legates. And so negotiations happened, piecemeal and gradually, and one by one over years the issues were settled.
It's notable that Henry II had no particular problems with Archbishop Theobald, Thomas's predecessor, nor any particular problems with his successors.
And they lost Thomas, who had been the stone that had blocked the way for the church getting any concessions for years.
King Henry II might seem to have lost, but not so much. Young Henry II had been dogmatic about having exactly the same situation as on the day when his grandfather Henry I was alive and dead, not seizing one bit more but not accepting one bit less. It got him into plenty of trouble, never more than in this case: canon law and the attitudes of the church had changed, and the church was now against many practices that they'd accepted under Henry I. Older Henry II apparently realized the subtleties of Henry I's policies, and generally tried to achieve the purposes he wanted without insisting on the letter of the past.
He abjured Henry I's right to block appeals to Rome ... but got the church to agree that suspect appellors "shall give you security that they will not seek the injury of you or your kingdom", which is what really concerned him, and thereby got the concession that he had the right to know and to enter his own arguments.
A good illustration of how compromise helped both was papal legates. Henry I had insisted that all communications with Rome had to go through him, and no legates could enter without licence. In 1176, a legate started to transit England to get to Ireland and Scotland. Henry II sent bishops to ask why he landed and "to intimate that unless he was ready to abide by the will of the king he would not be allowed to proceed further". The legate was forced to swear that he would do nothing hostile to Henry or his realm, but Henry then gave him letters of protection and an escort. Warren wrote, "The cooperation of the Crown was convenient -- and not very costly". The next legate, in 1178, sought out royal licence and offered the oath.
The church won most strongly on the issue of criminous clerks: they were to be judged by the church only, and only on a few serious crimes (as defined by the church, like heresy) the secular courts would have no jurisdiction, and the church defined (in a changing way) who had benefit of clergy ... but proof had to be made in a secular court before a royal justice; if a royal justice wasn't there, the accused had to sit in jail for months or years; if the accused fled before this judgment, there was no assumption of benefit of clergy and he was outlawed as usual; the accused's chattels were seized by and held by the king, released only if he was acquitted; he might be forced to abjure the realm (to go into exile) anyway.
And Henry didn't lose even on principle in every case. In the Constitutions of Clarendon, he had insisted that all pleas of debt must be held in royal courts. Canon law had come to insist that all pleas of debt must be heard in clerical courts, because there was a pledge of faith, an oath. The compromise was that pleas concerning the debts of laymen belong to the Crown, and any pledged security was a separate issue from the punishment of breach of an oath (punished by the courts). Advowson was the right of a secular donor to appoint the holder of a benefice, and the church never gave up its insistence that church offices could be judged only by the church, but Henry II developed the writ of prohibition to block specific cases (but only on request of the defendant; if both parties wanted the issue in a church court, he didn't interfere), and church and secular authorities came to coöperate in practice.
And Henry II even came out the better on it in at least one case. Near the end of the great rebellion, in 1174, he went back to England and "barefoot and fasting submitted to scourging for the rash words which had caused the martyr's death". Very soon thereafter a party of Yorkshiremen had surprised a party of invading Scots and killed and captured them all, including capturing William I, king of Scots. The rebellion and invasions collapsed then. "There were some who attributed the unparalleled victory to the intervention of God at the mediation of St Thomas of Canterbury; but this too was sweet to King Henry's friends, for it effaced the stigma of the martyr's death".
And he got rid of Thomas, who had been the stone that had blocked the way for the king getting any settlement for years.
Thomas Becket may have been happy with the result too. Warren didn't think that he "necessarily ... craved the martyr's crown". But it should be noted that, when the monks of the cathedral tried to bar the doors, he had them opened, and William FitzStephen (his clerk and biographer) noted that he could still have escaped easily, down into the crypts, or through a door, up a stair, and into the attic. Thomas had always been grandiloquent, trying to elevate disputes about criminal jurisdictions into grand Gregorian disputes about the subjugation of the Church -- when he was summoned to a court, the archbishop^^(arrogantly) characterized it as "when Christ was judged in my person before the tribunal of the prince".
O.K., not everyone was happy. The king of France and other enemies had tried to make much of the Henry-Thomas dispute, and were not happy with Henry and the church going "grazie" "prego" kiss kiss.
I am no expert on church-crown relations in England thereafter, but I believe that there were generally smooth relations afterwards, and the church didn't fuss about advowsons or land inheritance (like how secular law had a different notion of legitimate birth than the church), and working out of compromises. Until Henry VIII, as aforesaid.