His web-shooters are mechanical devices filled with a fluid of his invention, the stupid stingers are gone, he can't communicate with spiders etc.
That's probably for the best. If Neil Gaiman's Anansi Boys
has taught me anything, it's that talking to spiders only leads to trouble, usually in the form of obnoxious, unwanted, long-lost relatives showing up and stealing your fiancee's virginity. Peter Parker's life is complicated enough without all that to worry about.