A quick note: I am talking about the item/artifact, not the raid. Also, I have not done a lot of research into this yet and just a theory I am coming up with off the top of my head.
What I know about the Heart of Azeroth: Magni Bronzebeard as the speaker for Azeroth gives it to the Champion; we get no idea about its creation as far as I know. Not only can it absorb Azerite, but unspecified “powers” too as demonstrated by the dragon aspects as you level up the Heart of Azeroth. There are rumors that the Old God/s may have corrupted the artifact in some way before it is given to the Champion(Keep in mind I said rumors, not facts). The Heart of Azeroth can be used as a weapon by firing energy at foes among other effects. It takes the shape of a golden disk with an added clear dome to house and show the Azerite inside.
What I know about the Dragon Soul: The item is entrusted to Thrall who was known as The World Shaman(Possibly similar to the Speaker title or a Champion?). We know the origin of the Dragon Soul, created by Neltharion(Deathwing) while he was tainted by the Old Gods. The item is able to absorb the power of the dragons and be used as a weapon to fire a beam of energy. It is shown to be a golden disk as well though with no added features.
Like I said, this is off the top of my head with little research, just enough to confirm a couple of points I bring up. My theory is that the Heart of Azeroth is either a new form of the Dragon Soul or even the old one repurposed into this new artifact. My reasoning beyond this is the similar design and uses as well as I don’t know if the Dragon Soul was ever destroyed. I can make an assumption that it was destroyed as we know that during the Cataclysm the Dragon Soul had to be pulled from the past(Well of Eternity instance) and used in our present before it was sent back to its own past.
I don’t know. It is all speculation and there is little empirical connection between the two, but it is a fun theory in my mind that I was excited to share. What does everyone else think? Any opinions, maybe evidence they wish to put forward to prove or disprove the theory. I am genuine curious if this is even plausible to the minds of others like it is for me.