I'd say the first to exist win, meaning the Mote of Hell magical darkness don't extinguish the light from Hallow, nor is the light from Hallow can illuminate the cloud created by Mote of Hell.
It looks to me like Mote of Hell "wins" this interaction. Although the word "darkness" with a small "d" is used within the description of the spell, it's not really a Darkness AoE. It is a cloud filled with other things that are described by the spell. As such, Hallow's Daylight clause's restriction that Magical Darkness can't extinguish the light does not apply, but the Mote of Hell restriction that magical light cannot illuminate the cloud does apply.
However, there isn't really anything within the Mote of Hell description to suggest that a spell like Hallow would be completely dispelled by some overlap. Keep in mind that the radius for the Hallow effect is much larger than the radius for Mote of Hell. So, I would rule that if there were any overlap, then the effect of Mote of Hell is in effect within that area of overlap but the effect of Hallow is not, but the remaining AoE of Hallow remains in effect.
This same interaction would apply between the spells of Daylight and Mote of Hell.
Darkness. Darkness creates a Heavily Obscured area. Characters face Darkness outdoors at night (even most moonlit nights), within the confines of an unlit dungeon, or in an area of magical Darkness.
VS
Darkness creates a heavily obscured area. Characters face darkness outdoors at night (even most moonlit nights), within the confines of an unlit dungeon or a subterranean vault, or in an area of magical darkness.
The point remains; Mote of Hell isn't really a darkness or magical darkness spell even though that is one of the words used to describe the overall AoE. The AoE for Mote of Hell is referred to as a "cloud" in its description, and that cloud is composed of a few different things, some of which are physical matter, which fill the space. This cloud is dark inside of it not because of any sort of darkness effect, but because the cloud is so thick with brimstone and other phenomena that it is just dark inside of it as a result. The light-based spells such as Hallow and Daylight do not prevent this from happening. They only prevent relatively weak darkness-based effects from working there, not physical clouds. But the reverse is different. The Mote of Hell spell does prevent light and magical light from illuminating the cloud. This interaction would be true regardless of the order of which spell was cast first.
That used to be the argument for Hunger of Hadar (2014), because "blackness" isn't really darkness. But in Hunger of Hadar (2024), they changed that specifically to make it clear it's actual Darkness.
Mote of Hell won't get the same treatment, since it's partnered content, but it doesn't even say "blackness" like Hunger of Hadar. It actually says "darkness"...
We can only work with the wording that we have for the spell. As written, the spell will create a cloud of fire and brimstone within a space that is affected by a Hallow/Daylight spell effect. It's the same sort of interaction that you would get if you cast a spell such as Fog Cloud or Stinking Cloud into that space. Nothing within the description of Hallow/Daylight spell would prevent such a cloud effect from being created there.
Since the Mote of Hell is a third-party creation as you've mentioned, they weren't particularly careful about precise wording -- instead of using the word "darkness" in its descriptive sense, they should have just paralleled the wording from Fog Cloud and Stinking Cloud, which both instead explicitly make those areas Heavily Obscured. It's the same intent mechanically, which is just to cause blindness to someone who is trying to see something there. It's a question of whether the spell creates a darkness effect, or is it just saying that the effect happens to be dark because of the effect which is created (the cloud of fire and brimstone)? In which case, it would have been more technically correct to use the Heavily Obscured terminology, but in my opinion that is the intent.
In addition to this, the spell description itself defines its own effect which is to cause a creature within the area to be blinded. This is similar to how the original Hunger of Hadar was written. Remember, Darkness or Heavily Obscured areas do not actually do this. They do not cause creatures within them to be actually blinded -- that's a common misinterpretation. So, in the case of Mote of Hell and Hunger of Hadar, they explicitly declare that creatures within these areas are indeed blinded.
So, in the end, perhaps a DM might declare that if this cloud is created within the space which is affected by the Hallow/Daylight spell then such a cloud is well-lit instead of dark. That would be mostly flavor with very little mechanical impact. But the cloud would still exist. It's not prevented or dispelled or otherwise non-functional. It still causes creatures within the cloud to be blinded since that's what that spell description says that it does. I suppose that creatures could see into such a cloud where they could not otherwise, so that's slightly different.
I would not rule it that way anyways. To me, the cloud is dark as a consequence of the fire and brimstone created by the spell, NOT because the spell is creating an area of Darkness. Very explicitly, "No light, even magical light, can illuminate the cloud".
Hallow says
But Mote of Hell says
So, is the cloud illuminated or is the magical light extinguished inside it? One of these two has to give in to the other, but which one?
I'd say the first to exist win, meaning the Mote of Hell magical darkness don't extinguish the light from Hallow, nor is the light from Hallow can illuminate the cloud created by Mote of Hell.
It looks to me like Mote of Hell "wins" this interaction. Although the word "darkness" with a small "d" is used within the description of the spell, it's not really a Darkness AoE. It is a cloud filled with other things that are described by the spell. As such, Hallow's Daylight clause's restriction that Magical Darkness can't extinguish the light does not apply, but the Mote of Hell restriction that magical light cannot illuminate the cloud does apply.
However, there isn't really anything within the Mote of Hell description to suggest that a spell like Hallow would be completely dispelled by some overlap. Keep in mind that the radius for the Hallow effect is much larger than the radius for Mote of Hell. So, I would rule that if there were any overlap, then the effect of Mote of Hell is in effect within that area of overlap but the effect of Hallow is not, but the remaining AoE of Hallow remains in effect.
This same interaction would apply between the spells of Daylight and Mote of Hell.
Prior spells like Mote of Hell or Maddening Darkness say darkness because capital D-arkness is 2024.
For example look no further than;
Darkness: For the duration, magical Darkness spreads from a point within range and fills a 15-foot-radius Sphere.
VS
Darkness: Magical darkness spreads from a point you choose within range to fill a 15-foot-radius sphere for the duration
The point remains; Mote of Hell isn't really a darkness or magical darkness spell even though that is one of the words used to describe the overall AoE. The AoE for Mote of Hell is referred to as a "cloud" in its description, and that cloud is composed of a few different things, some of which are physical matter, which fill the space. This cloud is dark inside of it not because of any sort of darkness effect, but because the cloud is so thick with brimstone and other phenomena that it is just dark inside of it as a result. The light-based spells such as Hallow and Daylight do not prevent this from happening. They only prevent relatively weak darkness-based effects from working there, not physical clouds. But the reverse is different. The Mote of Hell spell does prevent light and magical light from illuminating the cloud. This interaction would be true regardless of the order of which spell was cast first.
That used to be the argument for Hunger of Hadar (2014), because "blackness" isn't really darkness. But in Hunger of Hadar (2024), they changed that specifically to make it clear it's actual Darkness.
Mote of Hell won't get the same treatment, since it's partnered content, but it doesn't even say "blackness" like Hunger of Hadar. It actually says "darkness"...
We can only work with the wording that we have for the spell. As written, the spell will create a cloud of fire and brimstone within a space that is affected by a Hallow/Daylight spell effect. It's the same sort of interaction that you would get if you cast a spell such as Fog Cloud or Stinking Cloud into that space. Nothing within the description of Hallow/Daylight spell would prevent such a cloud effect from being created there.
Since the Mote of Hell is a third-party creation as you've mentioned, they weren't particularly careful about precise wording -- instead of using the word "darkness" in its descriptive sense, they should have just paralleled the wording from Fog Cloud and Stinking Cloud, which both instead explicitly make those areas Heavily Obscured. It's the same intent mechanically, which is just to cause blindness to someone who is trying to see something there. It's a question of whether the spell creates a darkness effect, or is it just saying that the effect happens to be dark because of the effect which is created (the cloud of fire and brimstone)? In which case, it would have been more technically correct to use the Heavily Obscured terminology, but in my opinion that is the intent.
In addition to this, the spell description itself defines its own effect which is to cause a creature within the area to be blinded. This is similar to how the original Hunger of Hadar was written. Remember, Darkness or Heavily Obscured areas do not actually do this. They do not cause creatures within them to be actually blinded -- that's a common misinterpretation. So, in the case of Mote of Hell and Hunger of Hadar, they explicitly declare that creatures within these areas are indeed blinded.
So, in the end, perhaps a DM might declare that if this cloud is created within the space which is affected by the Hallow/Daylight spell then such a cloud is well-lit instead of dark. That would be mostly flavor with very little mechanical impact. But the cloud would still exist. It's not prevented or dispelled or otherwise non-functional. It still causes creatures within the cloud to be blinded since that's what that spell description says that it does. I suppose that creatures could see into such a cloud where they could not otherwise, so that's slightly different.
I would not rule it that way anyways. To me, the cloud is dark as a consequence of the fire and brimstone created by the spell, NOT because the spell is creating an area of Darkness. Very explicitly, "No light, even magical light, can illuminate the cloud".
I would argue that mote of hell creates magical darkness and is a lower level spell.
The description of the Mote of Hell spell doesn't say anything about magical darkness.
It's darkness summoned and maintained by the spell. Unless the spell tells us otherwise, that will always be magical darkness.