Both for all intents and purposes and for all intensive purposes are widely used to mean “for all practical purposes” or “virtually.” But which one is correct? The standard idiom is for all intents and purposes, not for all intensive purposes, though if you were to say these two forms out loud it might be hard to tell the difference between the two.
The cause of the confusion is rooted in this phonetic similarity. For all intensive purposes is what is known as an eggcorn, a label invented in the early 2000s by linguist Geoffrey Pullum to describe words or phrases that are misheard and consequently reform into a new word or phrase. Like, for example, hearing someone say “acorn” and thinking they said “eggcorn,” which is how this phenomenon got its name. The difference is stark when written down, but when spoken, the two words sound very much alike. Unlike mondegreens, eggcorns generally retain the same meaning as the original form, as in the case of for all intensive purposes.
The nonstandard for all intensive purposes seems to have first appeared in the 1950s according to the Google Books Ngram Viewer, while a version of the standard idiom, “to all intents, constructions, and purposes,” is cited in the OED as early as 1546. Historically, the dominant form of the idiom was to all intents and purposes, but over the past several decades that has gradually waned as the “for” form has increased in popularity.
The number of linguistic features that change due to a simple mishearing are numerous. The word apron, for instance, used to begin with the letter n, but the audible difference between “a napron” and “an apron” is so infinitesimal that speakers removed the n entirely, assuming it was part of the word “an” rather than “napron.” The word orange has a similar story, as do scads of other words and phrases across numerous other languages.
All in all, right and wrong can be a bit fuzzy in language, but for now, for all intents and purposes, you should opt for the proper idiom over the eggcorn alternative. This is especially important in your writing, where the nonstandard for all intensive purposes cannot be misheard.