Is it as simple as "he didn't know what to do with it"?
"The Ring's primary power was control of the other Rings of Power and domination of the wills of their users. The Ring also conferred power to dominate the wills of other beings whether they were wearing Rings or not — but only in proportion to the user's native capacity. In the same way, it amplified any inherent power its owner possessed."
Wikipedia, with accurate citations.
As Andres F. said in his comment, Gollum's power level was too low to start with for the ring to have conferred much in the way of additional power.
I'd agree with what Nathan C. Tresch and Andres F. and say that he doesn't get significantly more powerful because he wasn't originally very powerful. However, I'd argue that Gollum is actually more "powerful" than you'd think. He has some rather remarkable abilities. He can climb head-first down a sheer cliff-wall, is a clever waterman, survive off seemingly little food (think about the Dead Marshes), and is surprisingly strong for his size. These could all be potentially attributed to his possession of the Ring for so long.
In some way Gollum/Smeagol was physically altered, enhanced in agility and stealth, he had surprising strength of body for his small stature and ability to catch with feet and walk on rock like a spider clinging to the surface (posibly changes that come from his adaptation to the underground environment induced by the Ring), but his mind was so consumed by it that he had no strength of will to control. The Ring was controlling him. Also for a long, long time (from the very moment he got hold of the Ring until he was captured on borders of Mordor) he didn't know what he had in his keeping, he thought only that this mere magic ring could grant invisibility and nothing more, he didn't realize it's full powers neither he claimed it for the very purpose of using that power, besides he would need years of training to properly learn how to use it if he had strength of will to begin with (as Galadriel said to Frodo ,,Before you could use that power you would need to become far stronger, and to train your will to the domination of others. Yet even so, as Ring-bearer and as one that has borne it on finger and seen that which is hidden, your sight has grown keener. You have perceived my thought more clearly than many that are accounted wise. You saw the Eye of him that holds the Seven and the Nine. And did you not see and recognize the ring upon my finger? Did you see my ring?” she asked turning again to Sam.").
Even when not wearing it both Sam and Frodo could somehow tap into it's power on occassion (also the Ring the closer it got to Mount Doom the stronger it became) and gain insight far greater than they could in other means (possibly due to power of understanding the Ring provides) it is quite possible that Gollum would be similar in this regard but at first as mentioned he did not know what he had and the Ring itself was less active (as it increased in activity once Sauron ,,sents out his thoughts" as it is in fact part of the will of Sauron himself and so can act on it's own but also Sauron has effect upon it from afar).
Much of the rings power is essentially tied up with the owner knowing that it is a ring of power, and willing to wield it to further their ambition in some way.
Thus, it's a bit of an exponential thing, the more power you have already (being a leader of men, a commander of an army, a wizard, whatever) and the more ambition you have to use it (to conquer lands, defeat enemies and so on), the more you might think it can assist you, and make you think you can control it such that you remain in charge of the power. We all know really that when used for such purposes, even if you believe you are using it for good purposes, it will soon ensnare you in some way.
Gollum had no idea what it was, his ambition was merely to own it for it's own sake, his wielding of it, and his ambition was pretty much limited to a bit of hunting and evading of the odd goblin. He had no grand scheme to further, and no knowledge (initially) that it could help further those aims if he did have them. If he had any inkling of wanting to, for example conquer the Misty Mountains and make the Goblins his slaves, or personal army, etc, then surely the ring would have initially seemed to facilitate that for him, as it stands because he was so devoid of such potential it really just proffered the most basic characteristics of extending his life and providing invisibility when worn.