umm ... uhhh ... actually, all the replies in this thread are spot on. Simple answer is that the two terms mean the same thing. But meanwhile, there's no one agreed-upon definition of what that "same thing" is. Any microphone manufacturer is free to design a "wide cardioid" according to their own idea of what that pattern ought to be, as long as it's somewhere in the expanse of turf between omni and cardioid. If "Puce Microphones" decides to introduce a "wide cardioid" that's only a tiny bit broader in pattern than a standard cardioid, nothing is written in stone that the rest of us could point to and say, "Nuh-uh!".
See, the thing is, the directional patterns that can be obtained from a single transducer form a continuous spectrum from omni through cardioid to figure-8. It's an analog situation--you can plant your flag anywhere you like. Cardioid is smack in the middle, for example (causing some people's eyes to light up as they think "Platonic ideal!" while a few cranks and grumblers say, "Fehh--neither fish nor fowl!"). But if you observe the actual microphones that are called "cardioid" you'll see a range of behaviors. The cardioid pattern has a precise trigonometric definition but any specific implementation will be ... what will be.
And that's even before we consider the question of "what is the directional pattern at different frequencies?"--since the uniformity or non-uniformity of the pattern at different frequencies is very different among different microphone designs, and that makes a HUGE difference to the way they pick up sound, particularly for live indoor recording (as opposed to close-miking in an acoustically dead studio, where most of the sound that matters arrives on axis, or most outdoor recording). No, we're NOT talking about that, even though it's maybe half of what really matters. No! (Trying ever so hard not to go there ...)(though I want to)(pant, pant)
Anyway, following that epic struggle, attached is a graphical sketch that I once proposed for a certain manufacturer's general catalog. It was intended to show where one of their capsules sits on the above-mentioned spectrum. Any microphone can be anywhere on this spectrum at any frequency in the audio range. But only a few places along the spectrum have definite names attached to them.
In the case of "wide cardioid" there's an embarrassment of riches: two competing names, both of which have major defects as items of terminology go. But "subcardioid" wins the prize for being truly obscure--most writers avoid it unless they hope to earn snob points simply by being in the minority.
Some microphones have even been designed to be continuously adjustable across the spectrum (including one vintage classic, the Neumann M 49), and about ten years ago one idealistic manufacturer introduced a system that let the user remotely set a pair of microphones to have different patterns in different frequency ranges. They could be (for example) wide cardioids at low frequencies, supercardioids in the midrange and figure eights up top if you wanted--and you could choose the boundaries of those frequency ranges. In terms of flexibility it was kind of a SoundField microphone on ster[e]oids, with only two channels of output but better-sounding capsules. You could even do the M/S-like trick of recording the signals "raw" and then making your choices later on in playback, in the comfort and privacy of your own home, bomb shelter or cardboard box. Unfortunately it also cost as much as a small herd of llamas, and not many systems were ever sold.
--best regards