It could do more. So could Sherman, the other notable tank driving around in 1942/43. However, there is an apples/oranges comparison involved here.
Tiger was a vehicle designed to do one thing, and one thing only. It was a breakthrough vehicle. Tough enough to punch a hole in enemy lines, which the medium tanks could then exploit. Tiger did that one thing quite well. At pretty much everything else, except defending a static position (which arguably tanks aren't best suited for to begin with), it was more a liability.
It was very expensive to produce, so you have fewer toys in your military toolkit to begin with. It was very difficult to get to the battlefields, requiring special preparation before getting on/off rail cars, and was too heavy to move any distance by truck, and too difficult to move by road due to wear and tear. Crossing rivers often required submerging and driving along the bottom simply due to the lack of capacity for many bridges to hold it. It was too slow and consumed too much fuel to partake in the exploitation phase. Maintenance was a total pain, even recovering a single tank took three large half-tracks all towing together. Man-hour requirements were significantly higher to keep the tanks running, man-hours which could be used for something else. And, for all that, it's not as if the tank was invulnerable.
On the other hand a medium tank was a general purpose war-winner. The Western Allies demonstrated that a heavy tank wasn't even a mandatory requirement, even if it made life easier in certain situations. In a war of maneuver, medium tanks will win out. In other words, Tiger might help win individual battles, but T-34 would help win wars.
Don't get me wrong, if you happened to be unfortunate to happen to find one of the few Tigers which were roaming around in 1943, you were going to have a day you would remember. But the chances of that were quite slim.
In his recent book Armoured Champion, Zaloga discusses two ways of deciding the "best" tank: what he calls Commander's Choice and Soldier's Choice. To sum it up, a soldier wants the biggest gun possible and the thickest armour possible, while the commander cares more about getting the tanks produced, shipped, supplied, maintained, etc.
The Tiger definitely had a bigger gun and thicker armour than the T-34, but these came at a cost:
The Tiger weighed about twice as much as the heaviest configuration of the T-34. The design had no room for an upgrade. The project to create a replacement for the Tiger began before Tigers have seen any combat, and modernizations to the design were very, very slight. Meanwhile, the T-34 was outfitted with a much larger turret and a roughly equivalent gun without much trouble.
The Tiger was a very expensive, hand-assembled machine. Broken Tigers were difficult to transport and repair. All of this meant that the amount of Tigers that could be produced and kept running was very small, certainly not enough to change the tide of the war. Meanwhile, the T-34 could be produced and kept running in enormous numbers. Since tank warfare isn't an honourable duel between two opponents, it was possible to take a lot of T-34s and send them to a section of the front where there weren't any Tigers or even any enemy tanks at all, break through, and then force the Tigers to expend their resources trying to play catch-up.
Much like the armour of the T-34, the armour of the Tiger was only impressive when it first came out. Both T-34s and Tigers could be knocked out from a considerable distance by existing anti-tank guns firing at their flanks and even the front armour was vulnerable to heavier guns (Flak 36 and 52-K respectively). While the purpose of the T-34 was to exploit breakthroughs and operate in situations where heavy guns were less common, the Tiger was a breakthrough tank, whose purpose was to go up against heavy guns in well prepared positions.