Zaner-Bloser, to my knowledge, is not so far off from Palmer except in its prescribed teaching style, as carried out over multiple years in a child's schooling. The final product, that is, looks quite similar, and the techniques are not so far off from one another.
D'Nealian was one of the final blows to nibbed pens in the U.S., even wiping them out of most Catholic schools: it was ballpoint from then on, almost without question. With longhand in the workplace largely supplanted by typewriters (and soon by personal computers), the public at large moved toward writing implements that were "easier to pick up." After all, few people have need for an instrument whose finickiness is made up for only by ease of use in long sittings. Ballpoints are among the easiest writing tools to care for; you basically keep them out of the wash. No ink refilling, no cleaning out converters between colors and brands of ink, no ink drying out after two weeks in the drawer, no worry of the dreaded "baby bottom nib" of new fountain pens. Buy a Bic, get on with life. There's much to recommend them. If your writing is purely utilitarian and miscellaneous—lists, thank-you notes, reminders, memos, etc.—you probably don't want to bother with anything more complicated.
As it was taught to me, D'Nealian seemed to be designed for easy entry, plain and simple. Many other methods tend to concentrate on technique early, quite apart from letter formation. In D'Nealian, students are taught the proper manner in which to hold a pen, but little else. And ballpoint pens don't really allow for the old "proper" grips that are still often taught, as other commenters have touched on. You have to exert force downward onto the page to get a ballpoint to write, while the old, loose tripod grip that is still recommended was designed for nibs that only needed to glance the page. So D'Nealian is associated in the minds of many students with cramping exertion, and forcing the pen into practice.
Anyway, as soon as you can hold a pen, you start tracing and copying letters. You basically "draw" them; the letter is taught, not the movement. That's about all I can remember.
I'm not in education, so my recommendation of one over another shouldn't be sought. But I would easily believe that there were great things to say about a system that gets children writing as soon as possible. This is particularly true if the manner in which we make our letters by hand will have no bearing on our future prospects. Which, of course, it almost certainly won't.
Personally, I think D'Nealian looks childish. And if you aren't writing with a flowing technique, I see little point for joining all letters in a word. Not that cursive has always been written with streamlined, flowing, efficient, "full-arm" movement, as in the Palmer method and some of its relations. But from my point of view, why waste the extra ink or graphite needed to join letters just to encourage cramp?
I don't think D'Nealian killed cursive; Palmer or similar methodologies, which are hellish to learn if you go by the book, would likely have done even worse. More or less everything that we deal with today is in manuscript, unless your apartment complex happens to have cursive lettering on its awning or something, in the hopes of looking nice. Or maybe if you're digging through old relatives' things. So it's natural for children not to care about it, and even resent the drills needed to learn it.
I think the argument that ballpoints killed cursive is naturally overstated. If ballpoints hadn't become popular around the same time that typewriters did, there may well still be need of different writing implements and quick cursive in the workplace. But certainly ballpoints do make cursive much more difficult.