I'm asking for those who love Yugoloths to showcase what is great and unique about them.
As a confirmed fan of both Planescape and 4E who likes the 'loths a lot, here's my shot:
First, it wasn't the gods that made mortals, it was the Primordials, who made the demons/yugoloths, too. I like the 4E Epic adventures take on this: the Primordials made the world but were about to destroy it (to remake it anew - the old life-death cycle) when the gods intervened (because they thought it was beautiful) with their servants (the angels). Now:
Devils are servants of the gods (angels) who have rebelled because "the Primordials are the real creators". They want to corrupt and collect mortal souls (to make them suitable to feed their rebellion and then use them to do so).
Demons are spawn of the primordials who hate the gods for overthrowing their progenitors; they want to inculcate hate in mortals to strengthen their forces with the mortal souls, eventually to overthrow the gods and destroy the universe (just as their progenitors had meant to happen).
Yugoloths are progeny of the Primordials, but, like mortals, they have decided that they are fine with continuing to exist. They despise the gods, though, for their genocide against their ancestors, and they despise mortals for sucking up to the divine impostors. Devils they find intriguing, but still divine impostors; demons they despise for wanting to end the existence they have come rather to value.
The 'loths intentions in the Blood War are actually simple: keep it going so that the demons and devils are at least somewhat distracted from their other objectives!
In addition, some 'loths might have found ways to contact imprisoned (generally more "benign", in the sense of not wanting what the demons want) Primordials and be serving their interests, as they were originally created to do.
But this is just one example: each (game) world could have its own schema for this sort of cosmic background.
Edit: in terms of game roles. 'loths could make good allies/foils against either demons or devils, or could be furthering the (obscure) plans of imprisoned Primordials in the world, something both demons and devils tend to be too tied up with their own ends to be good for.