In addition to his original powers, Spider-Man gains the following abilities after he fights the Queen in "Disassembled" (The Spectacular Spider-Man vol. 2 #15-20, 2004), and also after apparently dying at the hands of the vampire Morlun and being reborn ("Spider-Man: The Other," 2005). It is apparent that his powers gained from "The Other" are only available because a voice inside him says that he is "embracing the Other".
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Arthropod communication
After Disassembled, Spider-Man can mentally communicate with arthropods (at least insects and spiders), though he does not seem to be capable of controlling them like Ant-Man. He can sense their presence, or glance at any spider and instantly know what kind it is (Friendly Neighborhood Spider-Man #4). It is possible that this is an extension of his spider-sense. Spider-Man used this mental capacity to communicate with all of the Queen's followers, humans with an "insect gene," and learned how to deactivate a bomb (The Spectacular Spider-Man vol. 2 #20).
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Organic webbing
In "Disassembled," Spider-Man develops the ability to shoot organic webbing from his wrists, and he stopped using his Web-Shooters.
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Night vision
After "The Other," Spider-Man develops night vision, allowing him to see in the dark. (The Amazing Spider-Man #528)
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Eyes, teeth, and stingers
Overcome by rage in "The Other," Spider-Man's spider side overruled his human qualities. His eyes turn glowing red, and he develops fanged teeth, night vision, and sharp stingers, (Friendly Neighborhood Spider-Man #3). The stingers are within his arms, protruding from his wrists (he quips that the stingers are a form of "Logan envy" [issue # needed]). They are coated in a venom that inflicts temporary paralysis. Spider-Man cannot consciously control these stingers yet, so they are only triggered in overwhelming situations (Friendly Neighborhood Spider-Man #6).
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Spider mutation and cocoon
During "Disassembled," Spider-Man mutates into a giant spider, and then emerges from the spider's body with a new ability to communicate with arthropods and organic webbing.
In "The Other," Spider-Man seemingly dies and molts his dead body. He creates a cocoon, and emerges with all his bodily damage healed: lost teeth, broken bones, his lost eye (which Morlun had torn out), scars, and even lost tonsils. In the words of Tony Stark, Peter's "odometer's been reset." Since some species of spiders shed their skin once in a lifetime, this may be a one-time occurrence. (The Amazing Spider-Man #527, Friendly Neighborhood Spider-Man #4)