Reuven spoke to his father, saying: "You may kill my two sons if I do not bring him [Binyamin] to you. Deliver him to my hand, and I will bring him back to you. (42:37)
Yehudah said to Yisrael his father: "Send the youth with me, and let us rise and go, and we will live and not die, us and you and our children. I will be the guarantor for him [Binyamin], from my hand you may demand him, and if I do not bring him before you and display him before you, then I will be guilty before you forever." (43:8-9)
Yaakov rejects Reuven's offer, but accepts Yehudah's. What difference between the offers justifies Yaakov's differing reactions? Here are three possibilities. Only the first is well known, but the second and especially the third may be equally important.
1. What they said: There is much room for (speculative) analysis here, and I will just give one idea.
At first glance, Reuven's offer might seem better than Yehudah's. It is measure for measure: as punishment for losing his father's son, he will have to lose his own sons. His offer of two sons in return for Binyamin recalls the Torah's double punishment for theft, or else, alludes to Rachel's loss of two sons. All of this is evidence of Reuven's sensitivity and willingness to take on responsibility.
But Reuven's proposal also contains fatal flaws. Killing his sons would clearly be a case of two wrongs not equaling a right - Yaakov will hardly be appeased about the loss of a son by also losing grandsons. And, of course, the killing would be unjust because Reuven's sons have done nothing to deserve death. (Reuven could not offer that he himself be killed, because that would deprive Yaakov of yet another son - exactly what the offer is trying to avoid.) Capital punishment is appropriate in certain situations, but in return for losing Binyamin it is morally unacceptable. Reuven tries hard, but his offer severely misjudges the moral issues of the situation.
In contrast, Yehudah offers an moral consequence for losing Binyamin - that he be considered guilty, but that nobody should be killed as a result.
Perhaps Yehudah is alluding to Reuven's offer when he says "we will live and not die, us and you and our children". Unlike Reuven, he says, he does not contemplate any of Yaakov's grandsons ("our children") dying as a result of failure on the trip to Egypt. Having specifically rejected the problematic aspect of Reuven's offer, it's no wonder his own offer is accepted.
2. Who said it. When Reuven disgraced himself with his father's wife Bilhah (35:22), he likely lost not only his birthright, but his father's trust as well. Yehudah had no comparable scandal (the Tamar case was likely less serious, and anyway Yehudah took full responsibility for it in the end). So Yaakov was more willing to rely on Yehudah for such an important mission.
3. When they said it. When Reuven made his proposal, the brothers had just come back from Egypt and had plenty of food. Yaakov's rejection may have just meant that he intended to wait for as long as possible before returning to Egypt. Perhaps the famine would end and no trip would be necessary? But by the time Yehudah made his proposal, all the food had been eaten (43:2), and Yaakov had no alternative but to let the brothers go.