Yes and no. Specifically, unoffically, if a judge tells me to take a case, I do. Why? Cause he's a freaking judge.... I'm not disobeying the guy, right, wrong or indifferent. He can tell a baliff to toss my ass in jail. No thanks. I never worked for a dedicated "Public Defender's Ofiice," but rather was "Appointed Counsel." They are very similar, but with some key distinctions. I didn't get a steady paycheck for one and got allegedly paid by the case (underpaid or sometimes never approved for payment, some judges have been indicted based on this and other factors. Still doesn't get me paid though).
Essentially, there is little difference to the individual criminal defendant. I was essentailly freelance which is why I could take other types of cases and business clients for example. One of my many problems was that I look young and like a patsy to some people. Nopes.... Anyhow it is common practice to just call yourself a public defender to appoint clients, because they get it. If you say "appointed counsel" their response is "huh?" It's six of one .... Half dozen of the other....
I took an exceptionally large amount of pro bono cases. Probably several tens of thousands of dollars worth.... I was both rather accomplished in these cases and astonished at the amount of BS complaints filed against me and quickly thrown out. One woman tried to say I inappropreately touched her..... I am secretly transgender MtF and only like men.... That's right, for all intents and purposes, she tried to say the super duper gay touched her sexually..... No go, and as you can imagine, she had done this to other innocent people....
All of the BS complaints have turned me off to the idea of taking pro bono cases. If you ever wonder why you an't find a lawyer, it's cause some jerks made the ones around you so tired of the BS complaints that it isn't worth it / is too risky and annoying. The problem is that there ARE shady as hell lawyers out there, but for each jerk crying wolf .... The legit complaints get lost in a title wave of BS.