Can someone explain to me why modern engines use valve overlap as opposed to haveing the intake and exhaust valves opened seperately?

 

So that when the intake valves are opened, the air will not only have atmospheric pressure pushing it in, but momentum from the air being blown out of the exhaust valve. This results in more airflow into the combustion chamber which equates to more power.

 

It's not just modern engines. Any race motor or old 60's muscle car with a "lope" or rumpa rumpa at idle has a bunch of overlap. Larger amounts of overlap will generally give more power on the top end with some loss on the bottom end. Very little or no overlap will give great bottom end but will go flat real early. A good example is the GM TPI motors from 85-90ish Vettes. Great power right from idle, but dead by 4500. Some of that is also the intake design, but they were made to work together.