How did Judas die?

While it is true that these verses are of the same event, the question is why this portrayal? The answer explains to so-called contradiction.

Judas hung himself (Mt. 27:5). He fell and split with bowels gushing out (Acts 1:18). What's it mean? The answer begins in Deuteronomy.

And if a man have committed a sin worthy of death, and he be to be put to death, and thou hang him on a tree: His body shall not remain all night upon the tree, but thou shalt in any wise bury him that day; (for he that is hanged is accursed of God;) that thy land be not defiled, which the LORD thy God giveth thee for an inheritance. Deut. 21:22-23

The chief priests agreed Judas had sinned against an innocent man; it was the price of blood (Mt. 27:5). That was a sin worthy of death. He hung himself, which would be from a tree. But rather than cut him down and bury him properly, they left him, until the rope rotted and Judas fell. They called it the field of blood.

I think I can add some useful informations in the subject of Judas death. Except of what is recorded in the New Testament, there are also other accoutns which may clear up potential consufion, remove contradictions and even propose entirely new view on the whole case of Judas death. In my opinion, they contain answer to question posted by OP: Are these two passages contradictory? How did Judas die? The answer is: no, passages are not contradictory - they just lack additional information "between". In the end Judas might have died in a way described in Acts, after he tried hanging itself but was rescued.

Let's start with quote from Papias, Apostolic Father, "hearer of John", which may initially add to general confusion about death of Judas but we will deal with it quickly:

Judas walked about in this world a sad example of impiety; for his body having swollen to such an extent that he could not pass where a chariot could pass easily, he was crushed by the chariot, so that his bowels gushed out.

Quoted from: http://www.earlychristianwritings.com/text/papias.html

Now it seems, that there may be a third option for Judas death: "crushed by the chariot". Other writers tried to explain it all and clear contradictions.

Quote from Apollinarius of Laodicea, 4th century bishop of Laodicea, who is also quoting Papias (or so he says):

Judas did not die by hanging, but lived on, having been cut down before he was suffocated. And the acts of the apostles show this, that falling head long he burst asunder in the midst, and all his bowels gushed out. This fact is related more clearly by Papias, the disciple of John, and the fourth book of the Expositions of the Oracles of the Lord as follows:

Judas walked about in this world a terrible example of impiety; his flesh swollen to such an extent that, where hay wagon can pass with ease, he was not able to pass, no, not even the mass of his head merely. They say that his eyelids swelled to such an extent that he could not see the light at all, while as for his eyes they were not visible even by a physician looking through an instrument, so far have they sunk from the surface.

His genitals appeared entirely disfigured, nauseous and large. When he carried himself about discharge and worms flowed from his entire body through his private areas only, on account of his outrages. After many agonies and punishments, he died in his own place. And on account of this the place is desolate and uninhabited even now. And to this day no one is able to go by that place, except if they block their noses with their hands. Such judgment was spread through his body and upon the earth.

Quoted from: http://www.chronicon.net/index.php/papias

Now quote from St Ephrem:

(...) when the rope broke, he fell and burst asunder (...) others say that Judas shut the door and barred himself in, and no one opened the door to see what was inside until his body was decomposed and all his bowels had gushed out

Quoted from: http://www.jstor.org/stable/3152829?seq=5

There is also very interesing quote from Theophylact (don't know which one, unfortunately):

Some say that Judas being covetous, supposed that he could both make money by betraying Christ and yet Christ not be killed, but escape from the Jews as he often did escape. But when he saw him now condemned and judged to die, he repented because the affaif had turned out other than he supposed it would. And this was why he hanged himself in order that he might get to hades before Jesus, and there implore him and obtain salvation. You must know however, that he actually put his neck into the noos, having hanged himself on a certain tree; but the tree bent down and he continued to live, because it was God's will either to reserve him for reprentance of for open disgrace and shame. For they say that he head the dropsy, so that he could hardly pass where a carriage easily could pass and then he fell on face and burst asunder.

http://www.jstor.org/stable/3152829?seq=6

Quote from Isho'dad, living in 9th century AD:

"He fell upon his face on the earth, and he burst asunder". They say that when Judas hanged himself either the halter was released and he escaped, or else someone saw him hanginng and saved him; and this happened by providence of God, first that the disciples might not be accused of having hanged him, and them because it ws fitting that he who had betrayed him openly should die openly. So he lived on and saw the resurrection of his Lord, and heard that he had come to his disciples many times, and that he had ascended to heaven; and then he came when many were gathered together and fell on the ground in the midst of the city, and burst asunder"

http://www.jstor.org/stable/3152829?seq=7

Qoute from Dionysius Bar Salabi who also quotes Papias and Epiphanius (I don't know which one):

"He went and hanged himself." Mathew sayeth this, but Luke in the Acts writes that he "burst in sunder" (...) and both are in the Right: (...) for after he (...) cast a Rope about his own neck in a Wood belonging to his House; and it happening that some passing by saw him hanging, and loosed him before he was choked. Others say the Rope broke, and that for some days after he was sick, and swelled to so large dimensions as that a cart could not bear him, and his head was sore puffed up and his eyelids so swollen that he could nod see. And Papias saith, that his privy members were mightily enlarged, and that putrid matter, abominable stench and Worms proceeded from them. Epiphanius saith, That he lived four days after his Suspension and that he was cut in twain and that his Bowels hushed out. Others [say] that he died of that Disease, and they did not bury him, for that i was a custom to leave those unburied who hanged themselves; Wherefore he did stink and became offensive, and a nuisance to the Inhabitants round about, and they were forced to remove him thence on a Bier; when they lifted him up he fell, and bursted and all his bowels gushed out. It is said by St. Luke in the acts of apostles, "Let his habitation be waste": That is to say, after they had buried him, the ill savour of his house offended the inhabitants, and they removed thence the stones and the rest of the materials, and so his habitation became waste, to wit, Scariot, and uninhabited. His houde was seated in Jerusalem.

Quoted from http://www.jstor.org/stable/3152829?seq=12

I do recommend for further reading: "Did Judas Really Commit Suicide?" by J. Rendel Harris

It's quite possible that both happened: he hanged himself, and when he was found and cut down, (which might have been some time later, long enough for the decay process to begin,) his body burst open with a predictable display of gore.