I was just in San Antonio - and you can't go wrong with either. That said, the Riverwalk area is amazing and it so great to walk around on. RiverCenter is a mall - and not a very good one at that. It is convenient to the Alamo - but also all the pricey touristy stuff - Ripley's, etc. Two place you MUST go, but will need a car. The Grist Mill in Gruene: www.gristmillrestaurant.com. The second is Texas Pride BBQ: www.texaspridebbq.net. You will NOT be disappointed!
With 2 boys you might consider the JW Marriott Hill Country Resort. They have wonderful amenities. The pool area is fabulous and golf as well. Lots to do! River walk is only a 30 minute drive from there.
From a San Antonian - With 2 teenage boys I recommend the Marriott Rivercenter - it is a little newer than the Marriott Riverwalk, and is attached to the Rivercenter Mall - a walk out the door of the Mall hits the Alamo, Ripley's Believe it or Not, Riverwalk and other attractions. Inside the mall is also an Imax theatre, restaurants, a food court, etc. You could feel safe allowing the boys to go downstairs and explore the variety of surrondings while staying close. With that said, the Marriott Riverwalk is just across the street, can't really go wrong with either.
Marriott Rivercenter. My wife and I went in May of 2010 for our anniversary. The Rivercenter is connected to a Mall. The canal boats start right at the back entrance to the hotel. The room was fantastic. We pulled into the Riverwalk by accident when we first arrived, the lobby was nice, but my vote goes to the Rivercenter.
I vote for the JW! Even though it is a short drive to the city, it is worth it! You will still be able to go into SA and experience the city, but you will return to an amazing resort. As Platinum members, they treated us like royalty.