Basically what it says on the tin. Having read though some of the materials on the issue, I am baffled by how recklessly the word is used, given the consequences of such usage.
Pedophiles are the people with sexual attraction to prepubescent children. It doesn't matter whether they do or don't act on that attraction; in fact, many don't. It is a sexual interest/mental condition that cannot be reliably changed.
Child molesters, on the other hand, are not necessarily pedophiles - in fact, 50 to 75% of child molesters do not have pedophilic interest.
Both facts can be sourced from the respective Wikipedia article and more info can be found in respective research.
Why does this matter?
Because the current use of the word reinforces stigma around pedophilia and makes it less likely for people with pedophilic disorder to reach out for help for the fear they would be outed and treated the same as actual child abusers.
This, in turn, makes those in a vulnerable position more likely to cross the line and get into the category of child abusers instead of coming for help. Also, it heavily affects people who did nothing to deserve such treatment.
What should we do?
We should leave the word "pedophile" to the context in which it belongs, which is the mental health and sexuality spheres, and avoid using the term to describe sexual offenders against minors. At the very least, one would most likely be wrong. At most, one would participate in the cycle of child abuse.
Pedophiles are people who randomly happen to have an interest in children. They can't do much about it; they can remove the sexual element (but not romantic one) by chemical castration, but most of them also experience attraction to adults, which will also get cut off.
In any case, this will feed into frustration, which may have other negative consequences.
Killing all pedophiles is not only inhumane, but also unfeasible, since you cannot separate them from others. And this won't remove the trait of pedophilia from the population; for all we know, the nature of pedophilia is not purely genetic.
It's also a self-defeating strategy. If it means paedophiles can't come forward to their therapist for fear of facing severe consequences, they...won't come forward. And won't get treatment to help them deal without causing harm. And will thus end up more likely to cause harm.