This was my family of 4's (2 kids, 14, 10) first cruise, so we did it up big with a concierge level room. While significantly more expensive than the standard room, we chose concierge primarily so that we could get a cabana on Castaway Cay (which we did). Here are the pluses:
The concierge lounge is a nice, quiet place to slip away to for a quick coffee, snack or drink. Not that you can't get these things elsewhere on the ship, but it was nice being able to do it in a well-appointed, quiet environment where you don't have to stalk for a table or be surrounded by loud conversations.
The concierge deck was also a great place to view the fireworks. Nice outdoor couches and chairs, a fireplace, a private bar--you watch the fireworks in comfort, while looking over at the massive crowd huddled shoulder to shoulder watching the same display.
The stewards were fantastic at providing information and assistance throughout the trip. We didn't have any special needs or requests, but it was clear that they were standing by ready to do whatever we needed.
I can't even imagine doing the beach at Castaway Cay without the private cabana. Otherwise, it just feels like any overcrowded public beach, with chairs and towels inches apart. The cabanas, which are also tended to by the concierge stewards, offer a nice, private area for you and your family to enjoy the day at the beach.
There really aren't many other perks to concierge. I assume the room is nicer, and don't get me wrong, it was a very nice room. But, we had a shared door with the room next to us, and noise was a problem the whole cruise. We had to hear a shrill mother disciplining her young children (who went to bed well after midnight for some reason) as if she was in our room with us. It seriously detracted from whatever luxury feel our room offered.
There is no separate pool, no private excursions, nothing else that is reserved for concierge. Once you step out of your cabin, you're going to the same restaurants, shows, pools, shops, etc. as everyone else. I don't say this out of snobbery. It's just an expectation we had based on the significantly higher price paid.
In the end, we compared concierge to flying 1st class on a plane. It's really hard to equate the higher price to the better experience. The flight is still the same, even if your seat is a little bigger and the food a little better. But it's also really hard to imagine going back to economy class after you've done it once.