Yes, there is a difference. As far as I can see, when you're climbing, you go higher up so you use ropes, harnesses and the like. Bouldering is more technical and closer to the ground so you just hang on as long as you can and when you fall, it's not far and there's a very soft floor awaiting your arrival.
We all went to CPH Boulders for a spot of the latter on Saturday afternoon. It was intended as a dry run for J's birthday party, but as they won't accept kids birthday parties, it turned out to be simply a family trip. We all really enjoyed it, there were easy and hard routes and plenty of space for the kids to play between climbs (or problems as they are apparently called when you're bouldering)
CPH Boulders is in the process of moving into this new location so it was a bit of a building site, but there was a really friendly atmosphere and we'll be back again. One of the consequences of that last statement is that I just ordered a pair of climbing shoes today because I had the largest pair they rent out and found them crippling so I found a bargain pair online and they're on their way to me as I write this.