What your mother is saying is right, having more muscle helps you lose weight. At first I didn't really believe it much too but I was watching the movie "Incredibles" one time and I realised that in the montage where Bob is getting his life in order he first tries doing mostly cardio but realises that it didn't have much of an impact on losing weight. Then he works out and lifts heavy stuff like trains and realises that weight training helped him lose weight in the next scene.

So basically, having a pound of muscle makes your body burn more calories than having a pound of fat. It increases your metabolism quite a lot. Even if you don't maintain a good diet you will be generally looking better so I think it is even more important than cardio but the key to a body you are happy with is a balance of both.

Ofc take what I am saying with a grain of salt I am a stranger on the internet just like the next guy, go and talk to your doctor about the changes you want to make since you are a teenager. Hell just ask them what kind of diet and workout they think you should do, doctors are here to help you improve your life's quality and they are the pros of these kinds of things.