The differences between these words is actually pretty simple. せんたく refers to laundry and washing clothes only. While あらう refers to just about every kind of washing (hands, car, dog, etc...), including washing clothes.
A helpful thing to do is check the example sentences with a good dictionary like jisho.org, usually the differences become pretty obvious:
I added a mem with this info to the words to help you out. I also changed せんたく to mean 'washing (clothes)' for a little more clarity.