I'll start this with pointing out a difference between the semantics of "web" and "cobweb". The difference implies that a "web" is maintained, and a "cobweb" is abandoned. Comparing the two, you can notice a difference in both strength and stickiness. I personally have experienced webs that were surprisingly resilient to being broken - similar to plucking a leaf from a tree almost. Whereas entire cobwebs can be brushed away by hand. Additionally, while a cobweb might "stick" to you when you walk through one, you can simply brush it off, whereas with a web, you do need to actually pull it off. The adhesiveness is caused by fluid coating the web in places, which would "dry up" over time and lose its adhesiveness.
Now, in the comic series, Spiderman is mostly known for his mechanical web shooter, and his special formula for its rapid disintegration. However, after constantly having to deal with running out of ammo, Peter does actually develop organic shooters. Since this was actually developed after the initial creation of the mechanical web shooters, it likely follows a similar design, i.e. the quick disintegration style.
However, in the movies, Peter starts with this power. There is very little talk about this, other than the constant debate about whether or not it was a good idea, so there is little else other than assumption that can be mode here. However, it can be explained with several assumptions:
It acts exactly the same way as normal Spider webs. After a time, it simply disintegrates on its own. Out in the streets of New York, the wind, rain, and general weather would likely take its toll on the web. And since peter is not overly involved in creating large webs on a regular occurrence, a few strands of web he uses to swing around on are likely to get blown away and eventually degrade over time anyway.
It acts the same way as the manufactured web works. Since it's stated that Peter got all of his abilities from the Spider that bit him - his physique, his enhanced reflexes, his fixed eyesight, and web, it could be implied that the web does not work the same way that manufactured web works - disintegrating quickly over the space of a couple of hours.
The Sam Raimi series does have a few inconsistencies from the original ideal of what Spiderman's abilities are, as well as how his enemies work as well. E.g., Doc Oc, for example doesn't quite work the way he does in the movies. Otto Octavius is obsessed with proving himself superior to Spiderman, not affected by a Rogue AI in his nervous system whispering to him like the devil on his shoulder.
So it can be sorted of "hand-waved" that the web works in such a way that it is not a menace to the city's Janitorial agency.