Bilbo slew the Mirkwood spiders in large numbers. Sam had the same weapon, Sting, and by the time he battled her, he had more weapons training than Bilbo when he saved the dwarves in Mirkwood. Why wasn't Samwise able to kill Shelob, a solitary spider? Was she not just another giant spider?
Sting did pretty well against Shelob, all things considered - small, untrained Hobbits vs. ancient GIANT spider, much larger than the spiders of Mirkwood.
from Book IV, Ch. 9 Shelob's Lair:
Far and wide her lesser broods, bastards of the miserable mates, her own offspring, that she slew, spread from glen to glen, from the Ephel Dúath to the eastern hills, to Dol Guldur and the fastnesses of Mirkwood. But none could rival her, Shelob the Great, last child of Ungoliant to trouble the unhappy world.
Sam laughed grimly. ‘Cobwebs!’ he said. ‘Is that all? Cobwebs! But what a spider! Have at ’em, down with ’em!’
In a fury he hewed at them with his sword, but the thread that he struck did not break. It gave a little and then sprang back like a plucked bowstring, turning the blade and tossing up both sword and arm. Three times Sam struck with all his force, and at last one single cord of all the countless cords snapped and twisted, curling and whipping through the air. One end of it lashed Sam’s hand, and he cried out in pain, starting back and drawing his hand across his mouth.
‘That would not help us now,’ said Frodo. ‘Come! Let us see what Sting can do. It is an elven-blade. There were webs of horror in the dark ravines of Beleriand where it was forged. But you must be the guard and hold back the eyes. Here, take the star-glass. Do not be afraid. Hold it up and watch!’
Then Frodo stepped up to the great grey net, and hewed it with a wide sweeping stroke, drawing the bitter edge swiftly across a ladder of close-strung cords, and at once springing away. The blue-gleaming blade shore through them like a scythe through grass, and they leaped and writhed and then hung loose. A great rent was made.
Stroke after stroke he dealt, until at last all the web within his reach was shattered, and the upper portion blew and swayed like a loose veil in the incoming wind. The trap was broken.
Book IV, ch.10 The Choices of Master Samwise:
But Shelob was not as dragons are, no softer spot had she save only her eyes. Knobbed and pitted with corruption was her age-old hide, but ever thickened from within with layer on layer of evil growth. The blade scored it with a dreadful gash, but those hideous folds could not be pierced by any strength of men, not though Elf or Dwarf should forge the steel or the hand of Beren or of Turin wield it. She yielded to the stroke, and then heaved up the great bag of her belly high above Sam’s head. Poison frothed and bubbled from the wound. Now splaying her legs she drove her huge bulk down on him again. Too soon. For Sam still stood upon his feet, and dropping his own sword, with both hands he held the elven-blade point upwards, fending off that ghastly roof; and so Shelob, with the driving force of her own cruel will, with strength greater than any warrior’s hand, thrust herself upon a bitter spike. Deep, deep it pricked, as Sam was crushed slowly to the ground.
No such anguish had Shelob ever known, or dreamed of knowing, in all her long world of wickedness. Not the doughtiest soldier of old Gondor, nor the most savage Orc entrapped, had ever thus endured her, or set blade to her beloved flesh. A shudder went through her. Heaving up again, wrenching away from the pain, she bent her writhing limbs beneath her and sprang backwards in a convulsive leap.
Not too shabby, really.
Short answer: No, she was not just another giant spider. Wikipedia points out that she is much, much older than the spiders of Mirkwood, who are, in fact, her descendants.
Shelob is described as the last child of Ungoliant, and it's usually taken literally to mean that she is Ungoliant's direct daughter. Ungoliant was far more than just a spider herself, being described as a "primeval spirit of night" who existed from "before the world".
So, no, Shelob was definitely more powerful than the spiders of Mirkwood, and it's pretty impressive how well Sam was able to stand toe-to-leg with her.
Tolkien implies that the spiders of Mirkwood were Shelob's offspring and were not as large or formidable as Shelob herself:
Far and wide her lesser broods, bastards of miserable mates, her own offspring, that she slew, spread from glen to glen, from the Ephel Duath to the eastern hills, to Dol Guldur and the fastness of Mirkwood. But none could rival her, Shelob the Great, last child of Ungoliant to trouble the unhappy world. (TT, Shelob's Lair, p.332 (HM Collector's Ed. 1987).)
The implication is that the reason Bilbo was more successful fighting the spiders of Mirkwood than Sam fighting only Shelob was because Shelob was a greater and more powerful creature. And as @Joe L. explained in his answer, Sam did a pretty decent job.
Officially, the spiders in Mirkwood were Shelob's descendants. She was described as "an evil thing in spider form" and the offspring of an even more powerful spider creature named Ungoliant. The Mirkwood spiders were basically mortal creatures, like eight-legged orcs. But Shelob was more like an evil spirit in physical form.
Ungoliant was a demon in spider form, who poisoned the two trees of Valinor (Telperion, and Laurelin, respectively.) Shelob was the last of her direct offspring (yes, Tolkien describes Ungoliant as she.) whereas the spiders of Mirkwood, were the offspring of Shelob. Mirkwood spiders were smaller third generation iterations. Shelob was more close the the giant and terrible monstrosity who was a companion of Melkor himself, though herself much smaller and less powerful. Ungoliant did indeed exist before Sauron.