Before anyone says something like "But I can/can't kill Vivec/Almy even before/after the heart is destroyed!", here's a clarification. Once they are still able to draw divine power from the heart, the Tribunal have two levels of existance. I guess they have both their incorporeal form somewhere (in Aetherius? The immortal plane.) and their corporeal form on Mundus (> Nirn > Tamriel) at the same time. You can kill this corporeal form at any time; they don't become stronger or weaker in the game no matter what happens to the heart. (Although, lore-wise, they are growing weaker all the time. This isn't part of the gameplay though.) Once this avatar-on-earth is dead, they're still living elsewhere. On the other hand, when the heart is destroyed, I think they lose their incorporeality and only exist as mortals once again. When you then kill the corporeal forms, they cease to exist permanently.
Here's how Vivec says it in topic "to be a God".
It is a bit like being at once awake and asleep. Awake, I am here with you, thinking and talking. Asleep, I am very, very busy. Perhaps for other gods, the completely immortal ones, it is only like that being asleep. Out of time. Me, I exist at once inside of time and outside of it.
It's nice never being dead, too. When I die in the world of time, then I'm completely asleep. I'm very much aware that all I have to do is choose to wake. And I'm alive again. Many times I have very deliberately tried to wait patiently, a very long, long time before choosing to wake up. And no matter how long it feels like I wait, it always appears, when I wake up, that no time has passed at all. That is the god place. The place out of time, where everything is always happening, all at once.So, they lose their ability to be "asleep" the second the Heart of Lorkhan has been destroyed, because their divinity only depends on it, unlike with "real" Gods.
This topic requires a real lore digger to fully explain everything asked here IMO.
The way I see it, the Tribunal are opportunist mofos that leech divinity to truly achieve fame and gain control over ordinary people (mer).