The nerdy and not too technical answer is that white, brown, jasmine and basmati and others are Old World rices, and slightly different than "wild rice" which is new world, which is more like a grass seed than a rice. But it's more like rice than wheat, barley or corn or other things that are also technically rices. which are all kinds of grass...
Or maybe the difference is about $4/pound?
nutritionally, brown rice (and brown jasmine rice is available to buy) is better for you than white (more complex carbs / fiber) and wild rice is even better than that.
There's other differences among rices, depending on what your interested in you may have to ask a botanist or a nutritionalist.