2 things.
1. Yes, it does alter your voice. Sometimes the tendency is to forget all technique in the heat of the moment and just whack, and over time, your voice will start to become a little raspy. If you do it often enough you find that you sometimes auto-lapse into your "distorted" voice, if you will, when you push. If you want to find out what bad technique does to your voice, talk to me, I'll show you. >.<
2. If you try to push your voice to sound like someone else, you're going to damage it even faster. Your best bet is to practise breathing directly into your diaphragm, as some of the others have already mentioned. Once you get used to it you'll find that by simply forcing the air out of your diaphragm, you can control the warble and tone of your screams. Try for short, controlled grunts first, then as you become more familiar with the way it works for you, try to control the amount of breath you throw out so you can push longer. If your technique is right, you should feel minimal or negligible soreness in your throat.
It's not something that can be explained like this, you got to see it in person to understand how it works.