First of all Spider-man doesn't have Organic webs in its starting stage in comics. He got Organic web later in the Comics series.
In the Sam Raimi's trilogy of Spider-man he does not follow the real Spider-man comics story as it is and skipped the Artificial Web-shooters thing and even skipped the main characters like Gwen Stacy. So, you can say its director/writer's choice to choose which aspects they grasp from the original content or not.
Here is some description of his main powers from origin to now in main universe of Marvel Comics -
When Peter Parker was bitten by a lethally irradiated spider,
radioactive mutagenic enzymes in the spider's venom quickly caused
numerous body-wide changes. Immediately after the bite, he was granted
his original powers: primarily superhuman strength, reflexes, and
balance; the ability to cling tenaciously to most surfaces; and a
subconscious precognitive sense of danger, which he called a
Spider-Man's web-shooters were perhaps his most distinguishing trait,
after his costume. Peter had reasoned that a spider (even a human one)
needed a web. Since the radioactive spider-bite did not initially
grant him the power to spin webs, he had instead found a way to
produce them artificially. The wrist-mounted devices fire an adhesive
In the "Disassembled" storyline Parker undergoes a transformation that
results in the ability to produce organic web fluid from his wrists,
and is able to fire his webbing in much the same manner as his
artificial web-shooters. According to the new 2007 Spider-Man
handbook, Parker has grown spinnerets in his forearms that terminate
in small pores at the junction of his wrists. By pressing down with
his middle fingers to his palm, he causes the pores to open and the
spinnerets to eject the organic fluid with a force equal to or greater
than that of his web-shooters.
Click here for more details.
By the way Sam Raimi's Script is inspired of James Cameron scriptment, which took the idea of organic web-shooters for
Stevens's Failed Script. From 1985 their have been many script
written for the Spider-men . But James Cameron's Script got most
attention and become the base of 2002's film.[source]
But the 2012's The Amazing Spider-Man follows the similar path to the comics and they choices the artificial web-shooter for their movie to establish the Peter Parker as a genius scientist.