The 2010 Sweet survey indicates that the range for neuropsych private practice is 37k- 840k. I think the differences are pretty dramatic.
A neuropsychologist making $840k solely in private practice likely owns a large group practice, which an LCSW is capable of doing as well. I'd focus more on the bottom end of that range ($37k), which is going to be much more common than the $840k figure anyway and falls pretty much in line with the starting salary for an LCSW.
LCSW:
http://www.payscale.com/research/US/Job=Licensed_Clinical_Social_Worker_(LCSW)/Salary
neuropsychologist:
http://www.payscale.com/research/US/Job=Neuropsychologist/Salary
Given the cuts in mental healthcare reimbursement (this largely excludes med management) and the over complicated insurance and billing process that accompanies our nation's healthcare system, I don't think anyone is arguing that mental health professionals w/o prescriptive authority who are going into private practice are going to be making $$$ anytime soon.
The most common private practice setting is likely a group practice, where a provider is working alongside several other providers (doctoral level and mid-level) and a large cut of their hourly rate (which is most likely already reduced due to reduced insurance reimbursement rates) is taken out by the practice to support overhead cost, support staff, and profit/income for the practice and the owner of the practice. I can only imagine that in this setting it would be hard for any provider - doctoral level or mid-level - to ever make a six figure income, and keep in mind that a provider working in this setting would most likely be on the hook for their own retirement, insurance, time off, etc.
The owner of that practice is probably going to be Ms. $840k, and she could be an LCSW, PhD/PsyD, etc - I think that the practice usually uses a corporate/group NPI to bill for services, but I'm only a neophyte in regards to PP organizational structures.
I think that what services a provider offers, rather than what degree a provider has, will have a lot more to do with their take home income. Working with services that are not covered by insurance is going to be the best route for making a steady income in PP, which I will admit is arguably easier for PhDs/PsyDs given their training in assessment/evaluation (psychoeducational/forensic testing, substance abuse evaluations, etc). Mid-level providers can still engage in court ordered therapy, rehabilitation/counseling with sex offenders/substance abusers, couples counseling (is that generally covered by insurance?), which are all generally not covered by insurance and require an out of pocket payment.
I have no interest in pursuing a PP career per se, but I would like to maintain a small, specialized, cash only practice, where I do a handful of assessments (psychoed, forensic, worker's comp, substance abuse, etc.) every couple months, in addition to a more traditional, 9-5 job.