The Public Domain in the US means the copyright of something is owned by nobody, and cannot be owned by anybody.
Open source is a license model. The creator of the software holds the copyright, and grants others a license to use it, under some conditions. The differences are
Ownership. When something is in the Public Domain, nobody has intellectual property rights to it. When something is made available under some open source license, the creator keeps the copyright.
Terms. When something is in the public domain, there are no license terms to follow because there is no license. You are permitted to use the work whichever way you see fit without one. When something is licensed under an open source license, each license sets its own set of conditions, which can differ drastically.