They are the same. The court appointed counsel is a Public Defender, (PD). You posted under the category of "family law," put attorneys are typically appointed only in criminal matters.
What is the context of your son's case?
The Public Defender is an attorney appointed by the court to defend a person who can't afford to hire his or own attorney.
You added, in response to another answer, that your son is charged with Penal Code §245(a)(1), which is assault by force likely to cause great bodily injury. It is a "wobbler," which means the District Attorney may file it as a felony or a misdemeanor.
Moved to Criminal Defense.
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Both are appointed by the court meaning you don't pay them directly. The PD is an office of attorneys who take the bulk of the cases. Other attorenys are appointed and paid by the government when conflicts of interest exist or multiple defendants.
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Both are used to represent indigent defendants and both are court-appointed. However, in the common understanding of the difference, the Public Defender works within one office, that is the Sacramento County Public Defenders office. A court-appointed attorney is a private attorney that does work not only on a retained basis but also accepts court-appointment to represent those defendants that the Public Defender cannot (because of an overload or a conflict).
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Public defenders are court appointed. Most counties also have alternate defense panels which are appointed where the PDs have a conflict of interest.
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