We have two boys, 5 years old and 4 years old. We witnessed nearly immediate, negative changes in mood and behavior, with long-term aftershocks (on the range of days) whenever we would let them play games on an iPad. There is a similar effect from watching TV (or treating the iPad as just a TV), but it takes much longer to set in, so a typical session doesn't seem to have as strong of an effect. On the other hand, I let my 5-year-old play Nintendo 2DS games and it doesn't seem to have more of an effect than TV. So there is something unique about the interactive tablet that makes it particularly bad.
The behavior looks a lot like stimulant withdrawal. There's an extreme irritability, tiredness, and seeking and pleading behavior. They look exactly like I do when I get crazy notions of quitting caffeine.
I think it's the combination of interactivity and direct interface. Having objects respond to your touch feels like magic. It's sort of like the delight of going to a petting zoo that never has to get turned off. And the critters on the screen don't just sit there when you yell "move, pony, move!" They actually move at your command, diligently and flawlessly.
When we consider that regular toddler misbehavior and temper tantrums stem from their inability to communicate effectively to be able to express their feelings and desires and have agency in the world, the sudden appearance of an object like the iPad, providing them perfect agency, must feel like an amazing power trip. The path to addiction seems pretty clear.
I think where standard video games don't have quite the same effect is the hand-eye coordination aspect. There's a disconnect between the input modality and the action on the screen. There is a metaphor that has to be learned. There are precise timings that have to be muscle-memorized to keep Mario from falling down a whole. It's difficult and that's why the game is fun, mastering the difficulty is rewarding. But it also slows the child down and keeps them from mainlining the dopamine hit.
So my wife and I have a basic rule of thumb: if we ourselves wouldn't sit down to consume a particular piece of media--be it because it's too boring or simplistic or patronizing for our adult tastes--that's a pretty good signal that it's probably designed to be addictive to toddlers. We then don't let the kids have it. They get to have Friday Night Movie (which we would do ourselves) and Saturday Morning Cartoons (which is just Netflix, because we don't have broadcast TV). During the week, we play card games like Uno, or do arts and crafts, for entertainment. As long as we're doing that, they don't even bring TV, video games, iPads, etc. up.
Which is why it pisses me off so much that our kindergarten gave the kids Google Chrome Tablets with educational "games" for "homework". The first 3 months, the kids' homework was basic stuff like "draw a picture of your favorite thing" or "practice reading for 10 minutes". But this month, the entire schedule is "do this one app on the tablet". And in less than a week, we're already seeing the impact. Our oldest doesn't want to do anything but use his school-provided tablet. AAARGH! It makes me so angry I could just do something really stupid, like running for school board.