Keefe's systems are more active. Players are always moving their feet in the offensive zone to either chase pucks or find open spaces for passes. There is a sense of structure where certain guys are lined up in certain areas by design, but if the flow of the play breaks up that structure, they adapt on the fly. Now it does lead to mistakes, since if a guy tries to be too overly aggressive he could be caught out of position and it could lead to odd man breaks the other way, but more often than not it puts a lot of pressure on opposing teams to make plays and it usually leads to mistakes on the their end. Plus since people's feet are often moving, and he runs with a fast team, they can recover a lot easier than getting caught flat-footed.
Babcock is more rigid. Guys are in their spots and unless the puck is near them, they don't move from that spot. It's up to the player with or near the puck to make a play with it, and they usually try to get it to the driver/playmaker of the line (i.e. Marner, Matthews, etc.) to make something happen with it. If done properly, it usually means guys are not out of position and it makes it easy for those playmakers to know and predict where their teammates are, but that also means it is very easy for the opposing team to predict where they are too, and all they need to do is shut down the lanes they know the playmaker will try to take, or just smother the playmaker altogether. Plus because people's feet are not moving, more likely to be caught flat-footed if there is a quick break the other way.