I am also studying the language so take that into consideration.

From my understanding, 了解 is understanding something/someone through knowing the thing/person for a certain amount of time. It usually comes from familiarity with something. Think of the expression "know something like the back of my hand." The expression is usually used for something you are very familiar with and understand very well. That's why 了解 is often used with a person. For example, when you fail to understand your girlfriend's feelings, you would use 了解.

明白 on the other hand is more like a realization or a revelation. It is that "AHA!" understanding when you figure out something. So, when someone says something to you and goes "y'know what I mean?" Then you would use 明白 to reply. This is why 明白 is often used with understanding the solution to a problem, understanding the gist of what someone is trying to say, etc..

So, I hope you 明白了my explanation, and the more you study Chinese, the more you will 了解 the language.