based on Siach Shaul, p. 224-5

When Hashem saved the Israelites and killed the Egyptians with great miracles at Yam Suf, the angels wanted to sing His praises. Hashem said to them: "The works of My Hand are drowning in the sea, and you are saying praises before Me?!" (Sanhedrin 39b). This is what Hashem said to the angels, whereas Bnei Yisrael were allowed and praised for singing to Hashem. What is the difference between them?
The standard answer is that Bnei Yisrael were directly involved in the miracle. However, that is not a complete answer. After all, Chazal tell us that the singing of Az Yashir was not just acceptable under the circumstances, but is that which set Hashem’s throne in this world (Shemot Rabba 23:1). What made Bnei Yisrael’s singing so appreciated?
There were two opposite and contrary worlds in creation before man was created: the world of atzilut, which was the realm of the angels, and the animalistic world of material, which we would call our world. When Hashem conferred with the angels before creating man, the angels protested: "What is man that you should consider him?" (Tehillim 8:5; Sanhedrin 38b). What bothered them was that man is comprised of the two opposites within his being. Why should there be such confusion and difficulty within one being?
In Hashem’s kindness, He gave man something of Him – not only by creating us in "His form" (Bereishit 1:27), however exactly we understand that, but by making us capable of performing creation (Sanhedrin 65b). This happens specifically because we come from both worlds and by acting in accordance with one or the other, we push in the direction of one of them. While the two elements can live separate from each other without confrontation, in man there is constant war between the worlds, one which sets a tone in the universe. When there is an increase in spirituality, there is also an increase in its opposition, as Chazal say: "the greater the person, the greater is his evil inclination" (Sukka 52a). If within man’s essence sanctity reigns, outside him there will be greater opposition containing impurity. Eventually the tension will lead to a conclusive struggle, as we will find the judgment at the Mountain of Eisav at the end of days (see Ovadia 1:21).
The angels cannot conquer the world of materialism because they do not have contact with it. Man can. Bnei Yisrael demonstrated self-sacrifice when they followed the instructions (Shemot 14:15) to march into the sea. It is true that the Egyptians followed them in, but that was not through the self-sacrifice that motivated Bnei Yisrael. The major step that Bnei Yisrael took was a choice of good over evil in a special historic manner, one which impacted the whole of history, and in that way set Hashem’s throne.
At that time, it was appropriate for man to sing about the miracle and its impact on history. This was one that was very human in its scope, not one that related to the world and the song of the angels.