The earliest story Bibles are often just that, a collection of stories that aren't really tied together by any larger theme. This can be problematic if a child never comes to understand the larger ideas of the Scriptures because the stories, when told this way, only hold a moral meaning without anything more. Take for example, the Veggie Tales stories. An asparagus is faithful and takes out a giant pickle with a single rock. The moral of the story (and the theme of the show) is that with God's help, little people can do big things too. But the next time a story about David is presented, the character is a cucumber who wants an asparagus' rubber ducky. Don't get me wrong, Veggie Tales are fun and entertaining, but they are somewhat disconnected. They don't very well tell the larger story and often, the stories only come from the Old Testament (because a good portion of the Old Testament is naturally a collection of sub-stories). The Bible story collections found in children's Bibles are often like the Veggie Tales--they are fun but disconnected from the bigger picture of the Gospel. When looking for Bibles that are appropriate for kids under 3 or 4, this is usually the only option. But that's okay because those collection of stories are a good start.
With my eldest son, we started with a story Bible that was printed on cardboard. It was well illustrated and bright. Most importantly, it was durable (something to think about when selecting books for young boys). But cardboard books only go so far. One day, the boy seemed to want to digest the Word, literally, eating and destroying the book with the power of slobber. So his mother and I set out to get something new.
Eventually we selected three Bibles to work through with my boys. The first is The Big Bible Storybook (Candle Books) edited by Maggie Barfield. It contains 188 stories from the Bible and is reasonably well balanced between the Old Testament and the New. The illustrations are actually photographs of puppets. It's different and fun. Another great element is how this Bible has taken more than just the simple stories. This Bible has incorporated some of the Law, Psalms, Parables, and even stuff from the Epistles. The primary reason we selected this story Bible was for the brevity of each self-contained story. Each story is short, maybe only a paragraph, and printed on a single page, making it easier for a hyper 3-year-old to sit through a story or two.
But at some point, a child needs to start seeing the bigger picture of the complete biblical narrative. (I think there are even many adults that have never moved past the collection of disjointed stories, sadly.) The second Bible we selected is the The Big Picture Story Bible (Crossway) by David Helm. This Bible goes beyond the collection of popular stories and begins to sew the bigger story together. It's broken into sections to show God's creation of, and interaction with humanity from Genesis to Revelation; and each section has somewhat natural start and stopping points. A parent and child can easily work through a three or four page story together, and there's more "story" on each page. The illustrations are fascinating, drawn from a aerial perspective, almost as if the reader is looking in on the story. These are certainly not the Sunday-school pictures I grew up with.
And the third Bible (which I highly recommend) is phenomenal for children starting at about the Kindergarten age, or maybe a little younger. It's The Jesus Storybook Bible (Zondervan) by Sally Lloyd-Jones. The illustrations are spectacular and whimsical. Even at a young age, my son loves the pictures, following the lines with his fingers as if trying to physically connect with the artwork. And this children's Bible is all about the Gospel of Jesus Christ, from start to finish. It's not a collection of stories, but one big story, THE STORY. It's artistic, poetic, and engaging. The great thing about a Bible like this is its far reaching impact into a child's biblical knowledge and early theological development. This is a great step between a children's translation of the Bible and a children's story Bible. It comes highly recommended and makes a great addition to every kid's bookshelf. But, remember: the value comes not from just sitting on the shelf or in simply looking at the pictures, the value comes when parents read God's Word to their children.
*I have no material connection to any of these recommended children's story Bibles.