I haven't visited Amersfoort, but Burgers' is incredible, and is one of the consensus best zoos in Europe.
There is no driving involved, but there is a fair amount of walking, as the zoo is rather big (it is entirely worth it, however). There aren't too many playgrounds, but there are a few, if that is a requirement, and in terms of great animals, there is just about every big name animal (the likes of elephants, rhinos, giraffes, bears, lions, tigers, leopards and manatees to name just a few) as well as plenty of smaller, but rarer and interesting species (such as Screaming Piha or Splitfin Flashlightfish) if that is something of interest to you. And the way the animals are displayed is very memorable, with two rainforest-themed greenhouses, the largest 'indoor desert' in the world, and Europe's largest aquarium away from the sea (by volume of water).
I would strongly recommend a visit, but I will leave it to others to comment on whether Amersfoort is worth visiting.