The school - not the style - is the more important factor. But, yes - there is a difference between the styles, and no, they are not taught the same (even within a style, schools teach differently)
EDIT: Better organization? There can be. If you absolutely must be in the olympics, then, you need to find a school who is aligned with the WTF. As of a few weeks ago, that can generally be any WTF/Kukkiwon school, but if you hold any ranking in ITF (any of the 3 main ones), or ATA, then you can transfer to the US National Dan program so that you can compete in WTF events, and, ergo, the Olympics.
There are other Taekwondo organizations, most are the "kwan" fraternal organizations, where you can't currently use their certification to compete in WTF/Olympics, though that could change in the coming months.
As such, it generally no longer matters which organization you belong to, if you want to compete in the Olympics. That, therefore, levels the playing field in terms of which organization is better. I will say that the IOC thinks that WTF has done a superb job of standardizing the events at the Olympics, in terms of fairness, scoring capability, quality of refereeing and judging, and standards, although I think this has brought on other problems. As such, the WTF federation at the highest levels are extremely professional and serious about the Olympics.
If this is your aspiration (Olympic Taekwondo), then, WTF is the best organization.
However, if you are not looking for Olympics, and prefer a more traditional approach, WTF is definitely NOT for you. Self-defense is not in their curriculum, and any instruction you get is at the whim of the local school's instructor. But this is generally rare; statistically, you're better of somewhere else. ITF has a better reputation for this, particularly the ITF whose headquarters is based in North Korea. But good luck finding a school to learn in, lest you end up being executed by being blown 11 miles away from a 8" anti aircraft gun.
ATA is another animal. Their schools are widely seen as money-making belt mills, and quality is hard to come by - even if expensive. There are good schools, but because of the marketing strategy of the ATA organization, the odds are against anyone finding a quality school, if you are looking for good self-defense or serious competition.
As far as it matters if the organization is recognized worldwide, that depends on what you want to do. If you want to compete internationally, or coach internationally, or referee or judge internationally, then yes, it absolutely does matter. If these are not your intentions, then no, it does not matter.