Accepting changes that were made

[Elladan and Elrohir]

Reasons why Aragorn might have let Frodo go (off the top of my head):

1) The Ring would have worked on him and whoever else he took with him the longer they went, thus producing the Boromir effect all over again.

2) His first priority was to deliver Minas Tirith from certain destruction. That may sound unimportant in light of the possibility of the Ring returning to Sauron, but Aragorn's first responsibility was to his own people.

3) His main stated reason in the book and movie: There was still hope for Frodo and Sam if they were allowed to go alone. There was no hope for Merry and Pippin if they were taken to Isengard. Also, the knowledge Saruman would have gained from them would have been detrimental. There was also the fact that Merry and Pippin never would have met Treebeard and so Isengard would never have fallen, but Aragorn didn't know that.

4) Exigency. He had to make a quick decision, and he didn't have time to think through the finer points of the implications of such a decision, such as my first two points (though they may have crossed his mind). He questioned himself several times afterwards (apparently), but Gandalf later comforted him by saying that he made a quick decision, and it turned out to be the right one.

5) He did what he felt, deep down, was the right thing to do. Frodo had gotten safely away, Boromir had almost conveniently fallen in battle, Aragorn was responsible for the Company since Gandalf's fall, the clear path of pursuing the Uruk-Hai had been laid out before him -- all the signs pointed to the decision he made.

It could have gone either way. Ultimately, Aragorn went with his gut feeling, and it turned out to be right.