As the NYTimes.com blog The Lede explains:
Lede derives from an intentional misspelling of the word “lead” (“lede” rhymes with “breed”), which developed in the newspaper
industry to avoid confusion with the kind of metal used in printing
presses (“lead” rhymes with “bread”). The lede of a newspaper’s front
page is the main story; the lede of an article is the way it begins:
the statement of facts or the anecdote that the writer lays out to
lead readers into a story.
It's a term specific to journalism, the newspaper industry in particular.
Of the 11 instances of lede that show up in the Corpus of Contemporary American English that aren't someone's name (belonging to one Cliff Lede), all are journalistic references from newspapers or news magazines.
They both describe the same action, so you could use either. They have different nuances, both of which work against John's case:
Commit is in this sense almost always used of a misdeed. We do not "commit a brave rescue plan", we only commit crimes, murders, misdeeds, sins, misdemeanors, assaults, and so on. As such, it carries with it a suggestion of blame. This isn't the case with carry out which can apply to commendable deeds.
On the other hand, carry out carries with it a connotation of prior planning. We do not carry out impulsive thefts. We cannot be provoked into carrying out an assault.
Now, "malice aforethought" is often part of the legal distinction between murder and other forms of culpable homicide* like manslaughter, or between degrees of murder (just what distinctions are made vary with jurisdiction), and has a bearing on other crimes (including planning for some crimes being a crime in itself) or upon the sentence received. As such saying John carried out a crime, suggests he is more culpable than if he had committed it - because it was premeditated.
So, while as bare sentences the two are very close, and we would likely favour the first due to commit being more specifically related to crimes and other misdeeds, as part of a larger passage we might favour carry out to emphasise that he was completing a previously decided plan. This decision might be to underline a greater degree of culpability, or just because we had already described the plan, and were now moving on to talking about its execution.
*As distinct again from justifiable homicide such as in self-defence or war, which are not crimes.
Best Answer
There is no difference semantically. The only difference is that "suicide" as a verb is so rare I have never seen it before; whereas "commit suicide" is common. I would advise against unusual usages in general: You might confuse people, which is ill-advised, and in this case it is totally non-constructive to the language.