There is a new and very detailed study of this topic, here is the abstract:

The right to an equal and fair trial regardless of wealth is a hallmark of American jurisprudence. To ensure this right, the government pays attorneys to represent financially needy clients. In the U.S. federal court system, indigent defendants are represented by either public defenders who are salaried employees of the court or private attorneys, known as Criminal Justice Act (CJA) attorneys, who are compensated on an hourly basis. This study measures differences in performance of these types of attorneys and explores some potential causes for these differences. Exploiting the use of random case assignment between the two types of attorneys, an analysis of federal criminal case level data from 1997-2001 from 51 districts indicates that public defenders perform significantly better than CJA panel attorneys in terms of lower conviction rates and sentence lengths. An analysis of data from three districts linking attorney experience, wages, law school quality and average caseload suggests that these variables account for over half of the overall difference in performance. These systematic differences in performance disproportionately affect minority and immigrant communities and as such may constitute a civil rights violation under Title VI of the Civil Rights Act.

Here is a non-gated version of the paper. By the same author, here is a working paper on whether the certainty of arrest reduces domestic violence.