All public defenders are court-appointed attorneys, but not all court-appointed attorneys are public defenders.
In most counties, Public Defenders are full-time county employees. They are salaried lawyers whose job is defending people who can't afford to hire their own lawyers.
In some smaller counties --such as Butte County, where I practice -- public defenders work under contract to the county to provide defense services, but they are also free to take private cases.
In some cases, where the Public Defender has a conflict of interest -- say, they already represent a defendant in a multi-defendant case, or they have previously represented a witness -- the court has to go outside the public defender's office, and hire a lawyer at an hourly rate.
As with private lawyers, appointed attorneys vary widely in their ability and experience. Public defenders are usually proven attorneys with supervisors to keep track of their work and ongoing training; court-appointed attorneys who are not public defenders can be people who would like a full-time public defender job but can't get one... or they may be experienced lawyers who take court-appointed cases for far less than their usual rate as a public service.
Please understand that this is a general discussion of legal principles by a California lawyer and does not create an attorney/client relationship. It's impossible to give detailed, accurate advice based on a few sentences on a website (and you shouldn't provide too much specific information about your legal matter on a public forum like this site, anyway). You should always seek advice from an attorney licensed in your jurisdiction who can give you an informed opinion after reviewing all of the relevant information.
Legal Aide is an organization of lawyers specifically hired and paid by the county to represent anyone who cannot afford their own lawyer.
A Court appointed lawyer is someone who is normally a "street lawyer" meaning they have their own private practice and they take assigned cases when a court assigns them to represent someone who is indigent and may have a conflict with being represented by Legal Aide.
It is not up to the client to decide which type of lawyer he/she receives from the court. It is solely up to the court. In my experience, you can get both good and bad lawyers either way since every lawyer is unique. Those in Legal Aide are usually busier, but have more people backing them up than assigned lawyers. The assigned lawyers usually want to service you better becaue they hope to impress you enough to have you refer new clients to them that are not assigned so they can charge a normal street fee which is higher than assigned hourly rates.
You may call our office at 516-248-6600 or send an email to us at Ted@Thelawteam.com. This answer does not form an attorney/client relationship with anyone and any answers do not constitute direct legal advice and should not be followed unless and until you have spoken with an attorney of your choice.