In the anime, all Kumoko does is refuse Mother's (Mother is here referring to the Queen Taratect who birthed Kumoko) order to come back to the Great Elroe Labyrinth after finally escaping it. Is there some deep reason that spiders aren't supposed to leave the labyrinth or something, important enough that it would spark a conflict between Kumoko and Mother? It just seems really odd they would both expend so much effort on one spider leaving.
This confusion stems from differences in the anime and light novels
The source of their conflict is actually pretty straightforward, and this is yet again another change in the anime that muddles the story a bit. In the anime, Mother tries to control Kumoko using her Kin Control skill, trying to get her to come back to the labyrinth, but Kumoko is immune by this point, likely due to Heresy Nullification. Immediately after refusing the order, a spider hit squad is sent after Kumoko, which she wipes out, and the conflict thus begins.
Mother attacking straightaway in the anime like that definitely contributed to making me think there was something mysterious going on, like some kind of spider covenant to stay in the labyrinth or something along that nature, but it's really not the case (not that I can really rule out something like that exists, having only read up to volume 4). But there's one big difference in the light novels that really makes this less of a mystery, and that is...
Kumoko instigated this fight
In the anime, it is after the events I just mentioned that Kumoko begins her psychic attack on Mother, but in the light novels, she begins it immediately after discovering her psychic connection to Mother, which happens even before her fight with the earth dragon Araba. This happens in volume 3, and she mentions that it was Mother's influence that made her get irritated at dragons, and that Mother only directly controls spiders that are powerful, and thus she discovers the connection when Mother tries to control her. Thus, while it cannot be said Kumoko wasn't provoked into the attack, she is the one who instigated this fight, and everything Mother does from this point on is just for self-preservation, as Kumoko's psychic assault definitely will kill her given enough time.
Demon Lord's Motivations
The Demon Lord Ariel's actions are directly tied to Mother's assault as well. Mother is one of her descendants (probably a direct one). It's a bit unclear what their relationship is exactly, but given spiders participate in cannibalism of their kin, most likely she just thinks of Mother as a very powerful ally, and wants to preserve her for this reason (it's said there are only a handful of Queen Taratects in the world, but they're all considered threats humans could never defeat).
My assumption turned out to be wrong, as later volumes make it clear Ariel genuinely does care about her kin, even calling them her beloved children in some cases, and this is especially evident with regards to her puppet spiders and the lost Queen Taratect. She mourns the loss of her Queen Taratect, and even prays for her soul. Albeit, Ariel does still have the Kin Eater title.
Interestingly, Mother's Heresy Resistance is level 9 (see page 45 of volume 4), so it's unclear how well Ariel can actually control her, so it's possible Mother is more of a willing minion. Though there's actually a bit of evidence Ariel can control her, such as the passage below, so take my little theory there with a grain of salt. Whatever their relationship, Ariel tries to kill Kumoko to try and save Mother.
Further information on page 161 of volume 4:
Hang on--why would a demon lord come to kill me anyway?
No, I guess I already know.
The Demon Lord's species name was origin taratect.
The zenith of all taratects, who even Mother would have to obey.
Just like my Parallel Mind told me in that emergency contact, she's even stronger than Mother, the monster we thought was the strongest of all.
She must have gotten some kind of distress signal from Mother and come to eliminate me.
Well, I guess my victory was only a matter of time, so it makes sense that Mother would request backup.
Kumoko as instigator, again, with the Demon Lord
Volume 5 of the light novels actually shows the Demon Lord and Kumoko's first encounter from the perspective of the Demon Lord, on page 152. Ariel politely asks Kumoko to stop attacking the queen, but there's some miscommunication, since Kumoko can't understand the language of the world. In Kumoko's chapter, she tries to make gestures to indicate she can't understand, but Ariel interprets these as a refusal.
"Hi there. I'm your grandma."
I jokingly introduced myself, approaching in a friendly manner.
"Sorry to ask this right off the bat, but could you stop attacking the queen now, please?"
Hearing that, the creature titled its head, then waved a leg from side to side.
It must have tilted its head to imply that it didn't know why I would ask such a thing.
"I see. I suppose I have no choice then."
It's not that it didn't occur to me to try to bring the Ruler onto my team.
However, I judged that it was too dangerous to leave this thing to its own devices.
If it wasn't going to listen to me, I'd just have to crush it on the spot.
If there's still any doubt that there's some big reason spiders can't leave Elroe Labyrinth...
In volume 4, it's revealed that a Queen Taratect lives in the elven forest, outside the barrier (and is one of the main reasons they need the barrier). Another one was mentioned as having been used to destroy one of the human forts in the Demon Army's big offensive (that was probably in volume 2 or 3).
What about the anime?
Since things happen differntly in the anime, we probably have to say anime Mother's motivations are a bit different. A reasonable conclusion is that Kumoko disobeying her orders was irritating enough to her to send spiders to attack her. The psychic assault later definitely still comes into play to escalate their conflict and increase Mother's desperation to kill Kumoko. The Demon Lord's motivations probably can be said to be more or less the same.