Spider-Man is a jumping spider, not a web spider
Despite the fact that they always have a black widow or false widow bite him on the big screen, the spider that bit him is definitely a jumping spider based on his powers. Or rather, or on the powers he doesn't have.
Here's his powers that he got directly from the spider bite:
Strength, speed, and reaction time proportional to that of a spider(read. 170x strength)
The ability to stick to walls
A "sixth sense" danger sense
Here's what he didn't get:
Webs(They're mechanical!)
Black widows are web spiders, meaning that they hunt by making a web and then waiting for prey to fly into it. They then bundle it up to eat later, and they don't actually need to eat that much, since they're sedentary. They are also not that good at climbing, preferring to make their webs near the ground, nor are they able to react quickly when not on their web.
Jumping spiders are hunting spiders, meaning that they hunt by stalking then pouncing on their prey much like a cat. They use their silk as a tether, to allow them to reach prey that is otherwise inaccessible and stay in place while making a kill, never to actually secure prey.
They also have much faster reaction times thanks to their four eyes and less sedentary lifestyle.
In short, his powers(and fighting style!) are closer to a jumping spider's than a web spider's, even though Marvel insists on using web spiders.