The short answer: the person in the armor is more likely to survive the bomb then the unarmored one, but the results may differ due to the handcrafted nature of both armor and bombs.
Now, coming to the details. First thing, I need to note that the bomb you link to is not a fire bomb, but a primitive grenade. It depends not on the fire for the effect, but on the blast wave from the explosion and damage from casing fragments.
The most likely fuse mechanism is a timed fuse - i.e., there is something that burns with more or less predictable speed, until it reaches the explosive (most likely, black powder) at the core. Black powder starts burning too, and since it is enclosed, the resulting gases expand and break the cast iron shell.
So, answering your question about what will happen, when the grenade hits the armor is this - it bounces and rolls, and continues rolling, until the fuse reaches the core. Then it explodes. In theory, you may rig some chemical impact fuse, but it would be more complicated, expensive and less reliable.
Now coming to the effects of the shrapnel on the armor. I wasn't able to find any data on the kinetic energy of the black powder IEDs. The data on the black powder firearms is interesting as well. Wikipedia gives absurdely huge numbers like 1700 Joules to the projectiles of early arquebuses. This paper here (https://www.researchgate.net/publication/321442172_ANALYSIS_OF_BALLISTIC_CHARACTERISTICS_OF_16th_CENTURY_ARQUEBUSES_USED_IN_BATTLE_OF_PAVIA) gives much more believable 830J (and decreasing with distance) for 30g lead ball, and explores it's effect on the armor contemporary to it.
Now, your grenade doesn't fire lead balls, it fires cast iron fragments - lighter for the same size, having less initial energy, since they are not fired through the barrel, having less aerodynamic shape, so loosing the speed and energy more rapidly. I would estimate their energy at about 100-200 J, comparable to the muscle-powered weapons (I'm absolutely winging it with numbers by this point).
The average medieval armor is also not uniform thickens, it is thicker at the more expected directions of impact - from the top and from the front.
So, what happens when the grenade explodes near the guy in armor very much depends on how near it is and from which side. If he is unlucky enough to be directly on top, he is toast. The blast wave and the shrapnel shred him to pieces.
Inside the ~10m radius the fragments may or may not penetrate, depending on where they hit. Less so from the front (unless he is unlucky enough to get one in the eyeslit), more so from the back. Outside that radius the grenade would, most likely, have no effect on the armored targets.
There are also blast wave effects to calculate, on addition to the shrapnel damage, but I do not have enough data for it.