Note that everyone is already infected, as explained by k0pernikus on the Sci-Fi SE:

In the comics it is established that everyone already is infected, though the symptoms only show after you "die" unless the brain is damaged too much.

This was shown with

Shane. He died by gunshot, yet turned into a zombie while laying six feet under without having endured any zombie-bites. After Rick learned of that fact, he dug him up again and killed him for good.

This also applies to the TV-show:

There was a man, who committed suicide by hanging himself in a tree, and he turned into a zombie as well. Andrea had him killed out of mercy.

The finale of the second season also confirms this:

Daryl brings up Randall, explaining that he turned without being bitten. "We're all infected," Rick says solemnly. "At the CDC, Jenner told me. Whatever it is, we all carry it." The others are furious at Rick for keeping Jenner's secret, but Rick insists he had no way of knowing if it was true.

Why walker bites are so deadly is not stated, so one can only speculate, like Jeff on the Sci-Fi SE:

Firstly, human mouths are dirty. A bite from a normal human is likely to become infected in many cases even today, when we have over-the-counter disinfectants, bandages, etc on hand in almost all locations. Left untreated, this alone could be fatal. This is unlikely today, in modern society, but not nearly as unlikely if society has broken down and medical treatment is unavailable. This is discussed to some degree in The Zombie Survival Guide by Max Brooks, which is based (as is TWD) on Romero-style zombies.

Next there's the reanimation. The virus is dormant, as you say, until death. If the zombie has any form of saliva (which they seem to) or moisture in their mouth (without which verbalizing would be nearly impossible) their bites will almost certainly introduce the live, active virus into the bitten person's bloodstream.

TL;DR: Being bitten by a zombie doesn't turn you into a zombie (it just kills you, and then something else, which you were already infected with, turns you into a zombie). As such, neither bites nor cuts from infected weapons have anything to do with why a person becomes a zombie.

Robert Kirkman, the creator of the Walking Dead franchise, has explicitly stated that "zombieism" has nothing to do with people dying of zombie bites.

The rule is WHATEVER it is that causes the zombies, is something everyone already has. If you stub your toe, get an infection and die, you turn into a zombie, UNLESS your brain is damaged. If someone shoots you in the head and you die, you're dead. A zombie bite kills you because of infection, or blood loss, not because of the zombie "virus."
- Originally written in the "Letter Hacks" section of an issue of the comic books, and quoted from the TWD wikia.

Being bitten kills due to either blood loss or mundane, non-zombie infection (e.g., sepsis). You turn, regardless of how you died, because the zombie virus was already inside you.

However, in the comics, the rules may be a bit different, and we have actually seen bad guys contaminate knives and arrows before attacking the good guys.

Possible spoiler, definite NSFW language:

You all know how this shit works. You get a bite, you get any kind of wound from these things, something from them gets in you...and you fucking die. — Negan, referring to zombies in the comics, quoted on the TWD wikia.

Wounds inflicted with these contaminated weapons do indeed prove fatal in the comics, although - again - they don't cause people to turn. The virus that turns dead people into zombies is already in everyone, on the show and in the comics.

So your question has now changed slightly:

Why do wounds from contaminated weapons kill people in the comics, but not on the show?

The only case in which this clearly happened on the show, to the best of my recollection, was Rick cutting his hand with his knife, immediately after using it to kill a zombie. This happened in season 6, episode 3, "Thank You". As of this writing, only a few hours (at most) have passed since that happened - according to Talking Dead, from the opening scene of the season, at the quarry, until the midseason finale, only one day has passed (the flashback episode notwithstanding). Thus, we don't know yet whether Rick will suffer some sort of complications from this wound, and since the next episode is the midseason finale, and will also take place during the same 24-48 as the previous episodes, we probably won't know until the midseason premiere in February.

We did see Tyreese having his arm lopped off with Michonne's katana in season 5, but in that case, he did indeed die shortly afterwards. However, he probably died from loss of blood rather than infection (there was so much fake blood used in those scenes that the crew had to replace the carpet on set several times), and if infection was involved, it was probably caused by being bitten twice, then having a significant amount of time pass before the bitten limb was amputated.

The Zombie virus is a two stage infection. The first stage has already infected everybody, but it is dormant or held back by an active immune system. This first stage is unable to kill a healthy human.

The second stage happens after death. When the immune system is no longer active the first stage can mutate and take over the corpse. This happens regardless of cause of death, unless the cerebellum is destroyed.

But introducing a second stage infection in a human will kick start the transformation. Being bit is almost guaranteed way of dying. Along with any other infection from the zombies mouth (less likely for the first wave of zombies), the immune system is compromised by the active zombie virus.

The simplest reason that an infected weapon won't infect a person is that the second stage virus, which is blood-borne, will not last long outside of its natural infection environment. Real viruses like the Cold (rhinoviruses) or Flu (parainfluenza) can only survive hours outside of the body, and infection rates decrease dramatically every hour. On a hard nonporous surface of a sword, your looking at few hours at best, which unlike poison, is just not viable for a weapon. It's not a good infection vector.

In comparison, snake venom also breakdowns within hours, while poison ivy oil will still be good for five years after secretion.