As a general rule, the US will not allow you to have multiple visas of the same type running concurrently.
However all this means is that once your new B1/B2 visa is approved, they will cancel your existing visa by stamping over it with "Canceled Without Prejudice". The "Without Prejudice" here refers to the fact that the visa wasn't canceled due to (for example) you breaching the conditions, but simply because they issued a new overlapping visa.
You new visa will start on the day of issue, and thus will cover the period of your trip.
Technically you don't need to apply for a new visa for the trip you've described. Unlike many other countries, the expiry date on a US visa is actually the date that you need to ENTER the country, not the date you need to leave - thus it would actually be possible to use your existing visa to enter on the August 14, at which point you would be given "status" that would allow you to stay within the US for (normally) 6 months. It doesn't matter that your visa would have expired by that time - as long as you enter the country by the expiry date you're good.
That said, I would NEVER plan to enter the country on the last day of a visa's validity. All it would take is a flight delay/cancellation and you'll find yourself in a situation where you will be entering after the 14th. In that case, you can expect to be denied boarding for your (rescheduled) flight on the grounds that you don't have a valid visa, or potentially refused entry into the US if you do happen to make it there.