Mike... Welcome to the Forum,
You are actually asking a couple of questions... they are both good.
The difference between an "AWS Certified" and a welder "Certified to AWS Standards" is that the AWS Certified welder had the test done at an accredited test facillity. When you have your tests at an ATF and pass (and pay yet another fee) your name will be posted on a national registry (for a short while) on an AWS website (that just about nobody reads)
The welder performance qualification test record "certified to AWS standards" has gone through the exact same rigor of inspection and testing as the welder performance qualification test record from an ATF.
Stork is one of the biggest and most respected testing labs on the face of the earth... Nobody has a better reputation. The paper you carry is every bit as valid as paper from an accredited test facillity.
I don't know personally of anybody who requires welders to be tested at an AWS accredited test facillity.
An accredited test facillity must go through an audit by the AWS to make sure they have the right equipment, That the staff has sufficient training and certification, and that a quality program is in place at the facillity. It costs money for this audit and frankly, much of industry doesn't care if a lab is an ATF or if it isn't... There is not a single ATF in the state I live in or the state that borders us to the west, yet plenty of welders get certified to the highest standards.