Those two should be distinguished also in Japanese.
The first sentence means something like "He intentionally avoided saying ～". This implies that saying it was usually expected, but he did not say it for a reason. The second sentence is "He did not dare to say ～". This implies he felt it was actually risky to say it, and he avoided it as a natural consequence.
We had to study hard for the final exam but he dared not to do so.
最終試験に向けて一生懸命勉強しないといけないのに、彼は敢えて勉強しなかった。 (Not studying was a risky thing for him, but he did not study, knowing the risk. Maybe he is a bad child.)
[X] 最終試験に向けて一生懸命勉強しないといけないのに、彼は敢えて勉強しようとはしなかった。 (He felt studying itself is useless/risky for him, and that's why he did not study. Maybe he knew he would fail anyway, maybe he was really sick, or maybe he had another important dream.)
We jumped from the roof one by one but he did not dare to do so.
我々は屋根から一人ずつ飛び降りた/飛び移ったが、彼は敢えてそうしようとはしなかった。 (Jumping from the roof itself was a risky thing at least for him. He was afraid.)
[X] 我々は屋根から一人ずつ飛び降りた/飛び移ったが、彼は敢えてそうしなかった。 (He was a ninja and knew he could safely do it, but he did not do so for his own reason. Maybe he thought jumping from the roof was not a good idea to accomplish his mission.)