As motivações do Aranha Negra estão detalhadas no Módulo de Campanha

Como a Lost Mines of Phandelver Adventure é disponibilizada gratuitamente pela Wizards of the Coast, vou citá-la aqui, mas vou colocá-la na marcação de spoiler para evitar estragar os jogadores que possam estar em uma campanha LMoP (não modificada).

Even though he intends to kill the characters, Nezznar is curious about them. Given the chance, he quizzes the characters at length regarding their identities, allegiances, interests, and goals. He files it all away in memory in the hope that someday he might find a use for what he learns.

Nezznar admits to being the Black Spider, and to using the Cragmaw goblins and the Redbrands to ensure that Wave Echo Cave remains his secret. He will say or do anything to put the characters off their guard, including promising to surrender or proposing cooperation against the monsters impeding his progress toward reaching the Forge of Spells. However, he betrays the characters as soon as they outlive their usefulness.

—Roleplaying [The Black Spider], The Lost Mines of Phandelver, Chapter 4

Há também algumas informações corroboradoras encontradas na introdução do módulo:

Unfortunately for the Rockseekers, they are not the only ones interested in Wave Echo Cave. A mysterious villain known as the Black Spider controls a network of bandit gangs and goblin tribes in the area, and his agents have followed the Rockseekers to their prize. Now the Black Spider wants Wave Echo Cave for himself, and he is taking steps to make sure no one else knows where it is.

—Background, The Lost Mines of Phandelver, Introduction

Abaixo está minha interpretação do que isso diz sobre suas motivações, o que pode ser óbvio no contexto, mas novamente na marcação de spoiler:

Essentially, Nezznar wants the Forge of Spells all to himself. It's a powerful magical landmark that facilitates the construction of magical items; it has intrinsic value in multiple ways. His motivations really aren't all that different from the Rockseekers, except that the campaign module frames his claim to the forge as being less legitimate than the Rockseekers. Regardless, he attacks the players because he views them as being a threat to his claim. As stated in the module, he doesn't attack immediately, and might even assist the party if the cave is still full of monsters, but he absolutely intends to betray them in order to keep the secret of the cave.

Being essentially the "Final Boss" of LMoP, he orchestrated Gundren's capture, and is more-or-less responsible for all the events of the campaign that occurred in reaction to the Rockseekers' expedition, including funding the Redbrands and the Cragmaw Goblins. So if the Adventurers rescued Gundren, they have good reason to view the Black Spider as an enemy, rather than as a mere annoyance or obstacle.